1. Fundamental Identities

Fundamental Trigonometric Identities

Trigonometric identities are mathematical equations that describe relationships between trigonometric functions. Unlike ordinary equations, which may only be true for particular values of a variable, trigonometric identities are true for every angle where both sides are defined.

These identities allow us to rewrite complicated expressions, solve trigonometric equations, verify mathematical relationships, and simplify calculations.

The three main groups of fundamental trigonometric identities are:

• Reciprocal identities: Relationships between trigonometric functions and their reciprocals.

• Quotient identities: Relationships expressing tangent and cotangent in terms of sine and cosine.

• Pythagorean identities: Relationships derived from the Pythagorean theorem and the unit circle.

Understanding these identities is essential for advanced trigonometry, calculus, physics, and engineering.

What Is a Trigonometric Identity?

A trigonometric identity is an equation involving trigonometric functions that is true for all values of the variable for which both sides are defined.

For example:

sin² θ + cos² θ = 1

This equation is true for every real angle θ.

At θ = 30°:

sin² 30° + cos² 30°

= (1/2)² + (√3/2)²

= 1/4 + 3/4

= 1

At θ = 45°:

sin² 45° + cos² 45°

= (√2/2)² + (√2/2)²

= 1/2 + 1/2

= 1

The relationship continues to hold for all angles.

Identities Compared with Equations

Consider:

sin θ = 1/2

This is a trigonometric equation. It is true only for certain angles.

However:

sin² θ + cos² θ = 1

is an identity because it is always true.

Identities are useful because they allow one expression to be replaced by another equivalent expression.

Reciprocal Identities

The reciprocal identities connect the six trigonometric functions.

The three fundamental reciprocal relationships are:

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

These can also be written as:

sin θ = 1/csc θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Each identity is valid wherever both sides are defined.

Reciprocal Identity Table

Worked Example: Using a Reciprocal Identity

Suppose:

sin θ = 3/5

Find csc θ.

Using:

csc θ = 1/sin θ

Substitute:

csc θ = 1/(3/5)

Therefore:

csc θ = 5/3

Worked Example: Finding Secant

Suppose:

cos θ = −4/7

Find sec θ.

sec θ = 1/cos θ

sec θ = 1/(−4/7)

sec θ = −7/4

The reciprocal has the same sign as the original nonzero value.

Understanding Reciprocal Relationships

Reciprocal identities follow directly from right-triangle trigonometry.

For an acute angle θ:

sin θ = opposite/hypotenuse

Taking the reciprocal gives:

1/sin θ = hypotenuse/opposite

This is the definition of cosecant.

Therefore:

csc θ = 1/sin θ

Similarly:

cos θ = adjacent/hypotenuse

Taking the reciprocal:

sec θ = hypotenuse/adjacent

And:

tan θ = opposite/adjacent

Taking the reciprocal:

cot θ = adjacent/opposite

Important Restrictions

Reciprocal identities involve division.

Division by zero is undefined.

Therefore:

• Cosecant is undefined when sine equals zero.

• Secant is undefined when cosine equals zero.

• Cotangent is undefined when sine equals zero.

For example:

csc 0° = 1/sin 0°

= 1/0

This is undefined.

Likewise:

sec 90° = 1/cos 90°

= 1/0

This is undefined.

The expression 1/tan θ also requires tangent to be defined and nonzero.

Quotient Identities

Quotient identities express tangent and cotangent using sine and cosine.

The two fundamental quotient identities are:

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

These identities are particularly useful because they allow expressions involving tangent and cotangent to be rewritten using only sine and cosine.

Deriving the Tangent Identity

From right-triangle trigonometry:

sin θ = opposite/hypotenuse

cos θ = adjacent/hypotenuse

Divide sine by cosine:

sin θ/cos θ

= (opposite/hypotenuse)/(adjacent/hypotenuse)

The hypotenuse factors cancel.

This leaves:

opposite/adjacent

which equals tangent.

Therefore:

tan θ = sin θ/cos θ

Deriving the Cotangent Identity

Similarly:

cos θ/sin θ

= (adjacent/hypotenuse)/(opposite/hypotenuse)

= adjacent/opposite

Therefore:

cot θ = cos θ/sin θ

Applying Quotient Identities

Worked Example: Finding Tangent

Suppose:

sin θ = 5/13

cos θ = 12/13

Find tan θ.

Use:

tan θ = sin θ/cos θ

Substitute:

tan θ = (5/13)/(12/13)

Simplify:

tan θ = 5/12

Worked Example: Finding Cotangent

Suppose:

sin θ = −3/5

cos θ = 4/5

Find cot θ.

cot θ = cos θ/sin θ

= (4/5)/(−3/5)

cot θ = −4/3

The negative result is consistent with an angle in Quadrant IV.

Simplifying an Expression

Simplify:

sin θ × cot θ

Replace cotangent using the quotient identity:

sin θ × (cos θ/sin θ)

Cancel sin θ:

= cos θ

This simplification is valid where sin θ ≠ 0.

The Fundamental Pythagorean Identity

The most important trigonometric identity is:

sin² θ + cos² θ = 1

This relationship comes directly from the unit circle.

The equation of the unit circle is:

x² + y² = 1

For a point on the unit circle:

x = cos θ

y = sin θ

Substituting gives:

cos² θ + sin² θ = 1

Therefore:

sin² θ + cos² θ = 1

Connection to the Pythagorean Theorem

For a right triangle:

a² + b² = c²

On the unit circle, the hypotenuse has length 1.

The horizontal and vertical components are cos θ and sin θ.

Therefore:

(cos θ)² + (sin θ)² = 1²

This produces the fundamental Pythagorean identity.

Rearranging the Pythagorean Identity

Starting with:

sin² θ + cos² θ = 1

We can rearrange to obtain:

sin² θ = 1 − cos² θ

and:

cos² θ = 1 − sin² θ

These forms are useful when one trigonometric value is known and another must be calculated.

Worked Example

Suppose:

sin θ = 3/5

and θ lies in Quadrant I.

Find cos θ.

Use:

sin² θ + cos² θ = 1

Substitute:

(3/5)² + cos² θ = 1

9/25 + cos² θ = 1

cos² θ = 16/25

Take the square root:

cos θ = ±4/5

Since θ lies in Quadrant I, cosine is positive.

Therefore:

cos θ = 4/5

Why the Quadrant Matters

When taking the square root of an equation, both positive and negative values may be possible.

The quadrant determines the correct sign.

For example, if the same sine value occurred in Quadrant II:

cos θ = −4/5

The Second Pythagorean Identity

Starting with:

sin² θ + cos² θ = 1

Divide every term by cos² θ:

sin² θ/cos² θ + cos² θ/cos² θ = 1/cos² θ

Using quotient and reciprocal identities:

sin² θ/cos² θ = tan² θ

cos² θ/cos² θ = 1

1/cos² θ = sec² θ

Therefore:

1 + tan² θ = sec² θ

This is the second Pythagorean identity.

It is valid where cos θ ≠ 0.

Rearranged Forms

tan² θ = sec² θ − 1

1 = sec² θ − tan² θ

Worked Example

Suppose:

tan θ = 2

Find sec² θ.

Use:

1 + tan² θ = sec² θ

Substitute:

1 + 2² = sec² θ

1 + 4 = sec² θ

sec² θ = 5

The Third Pythagorean Identity

Again, begin with:

sin² θ + cos² θ = 1

Divide every term by sin² θ:

sin² θ/sin² θ + cos² θ/sin² θ = 1/sin² θ

Simplify:

1 + cot² θ = csc² θ

Therefore:

1 + cot² θ = csc² θ

This is the third Pythagorean identity.

It is valid where sin θ ≠ 0.

Rearranged Forms

cot² θ = csc² θ − 1

1 = csc² θ − cot² θ

Worked Example

Suppose:

cot θ = 3

Find csc² θ.

Use:

1 + cot² θ = csc² θ

Substitute:

1 + 3² = csc² θ

1 + 9 = csc² θ

csc² θ = 10

Summary of the Fundamental Identities

These identities form the foundation for most trigonometric simplification and verification problems.

Verifying Trigonometric Identities

To verify a trigonometric identity, we show that the expressions on both sides are equivalent.

Unlike solving an equation, we are not trying to find a particular value of θ.

Instead, we demonstrate that the relationship is true for every angle in its domain.

A common strategy is to begin with the more complicated side and simplify it until it matches the other side.

Worked Example: Verifying a Reciprocal Identity

Verify:

sin θ × csc θ = 1

Start with the left-hand side:

sin θ × csc θ

Replace cosecant:

= sin θ × (1/sin θ)

Cancel sin θ:

= 1

Therefore:

sin θ × csc θ = 1

The identity is verified for sin θ ≠ 0.

Verifying Identities Using Quotient Relationships

Verify:

tan θ × cos θ = sin θ

Start with the left-hand side:

tan θ × cos θ

Replace tangent:

= (sin θ/cos θ) × cos θ

Cancel cos θ:

= sin θ

Therefore:

tan θ × cos θ = sin θ

The identity is verified wherever cos θ ≠ 0.

Another Example

Verify:

cot θ × sin θ = cos θ

Start with:

cot θ × sin θ

Replace cotangent:

= (cos θ/sin θ) × sin θ

Cancel sin θ:

= cos θ

The identity is verified wherever sin θ ≠ 0.

Verifying Identities Using Pythagorean Relationships

Worked Example

Verify:

1 − sin² θ = cos² θ

Start with the fundamental identity:

sin² θ + cos² θ = 1

Subtract sin² θ from both sides:

cos² θ = 1 − sin² θ

Therefore:

1 − sin² θ = cos² θ

Worked Example

Verify:

sec² θ − tan² θ = 1

Start with:

1 + tan² θ = sec² θ

Subtract tan² θ:

sec² θ − tan² θ = 1

The identity is verified wherever cos θ ≠ 0.

Verifying More Complex Identities

More complicated expressions often require several identities.

Worked Example

Verify:

(1 − cos² θ)/sin θ = sin θ

Start with the left-hand side:

(1 − cos² θ)/sin θ

Using:

1 − cos² θ = sin² θ

Substitute:

= sin² θ/sin θ

Cancel one factor of sin θ:

= sin θ

The identity is verified wherever sin θ ≠ 0.

Worked Example

Verify:

(1 + tan² θ)cos² θ = 1

Start with:

(1 + tan² θ)cos² θ

Use:

1 + tan² θ = sec² θ

Then:

= sec² θ × cos² θ

Replace secant:

= (1/cos² θ) × cos² θ

= 1

The identity is verified wherever cos θ ≠ 0.

Simplifying Trigonometric Expressions

Simplifying a trigonometric expression means rewriting it in an equivalent form that is easier to work with.

The simplest form depends on the purpose of the calculation.

Sometimes it is useful to express everything in sine and cosine.

At other times, a reciprocal or Pythagorean identity produces a shorter result.

Worked Example: Simplify tan θ × cot θ

Use:

cot θ = 1/tan θ

Then:

tan θ × cot θ

= tan θ × (1/tan θ)

= 1

This holds where tangent and cotangent are both defined.

Worked Example: Simplify sec² θ − 1

Use:

1 + tan² θ = sec² θ

Rearrange:

sec² θ − 1 = tan² θ

Worked Example: Simplify csc² θ − cot² θ

Use:

1 + cot² θ = csc² θ

Therefore:

csc² θ − cot² θ = 1

Simplifying Expressions with Fractions

Trigonometric fractions can often be simplified by replacing functions with sine and cosine.

Worked Example

Simplify:

tan θ/sec θ

Replace tangent and secant:

= (sin θ/cos θ)/(1/cos θ)

Multiply by the reciprocal:

= (sin θ/cos θ) × cos θ

= sin θ

This is valid where cos θ ≠ 0.

Worked Example

Simplify:

cot θ/csc θ

Replace both functions:

= (cos θ/sin θ)/(1/sin θ)

Multiply by the reciprocal:

= (cos θ/sin θ) × sin θ

= cos θ

This is valid where sin θ ≠ 0.

Using Algebraic Techniques with Identities

Many trigonometric expressions can be simplified using familiar algebraic techniques.

These include:

• Factoring

• Expanding brackets

• Finding common denominators

• Difference of two squares

• Cancelling common factors

Worked Example: Difference of Squares

Simplify:

(1 − cos θ)(1 + cos θ)

Use:

(a − b)(a + b) = a² − b²

Therefore:

(1 − cos θ)(1 + cos θ)

= 1 − cos² θ

Using the Pythagorean identity:

= sin² θ

Worked Example: Simplifying a Fraction

Simplify:

sin² θ/(1 − cos θ)

Use:

sin² θ = 1 − cos² θ

Then:

= (1 − cos² θ)/(1 − cos θ)

Factor the numerator:

= [(1 − cos θ)(1 + cos θ)]/(1 − cos θ)

Cancel the common factor:

= 1 + cos θ

This simplification is valid where 1 − cos θ ≠ 0.

Rationalising Trigonometric Expressions

Sometimes a trigonometric expression contains a denominator such as:

1 + cos θ

or:

1 − sin θ

Multiplying by the conjugate can simplify the expression.

Worked Example

Simplify:

sin θ/(1 + cos θ)

Multiply numerator and denominator by:

1 − cos θ

Then:

= sin θ(1 − cos θ)/[(1 + cos θ)(1 − cos θ)]

Use the difference of squares:

= sin θ(1 − cos θ)/(1 − cos² θ)

Using:

1 − cos² θ = sin² θ

We obtain:

= sin θ(1 − cos θ)/sin² θ

Cancel one factor of sine:

= (1 − cos θ)/sin θ

This is an equivalent form wherever the original expression is defined.

This technique is useful in advanced trigonometry and calculus.

Proving Identities Step by Step

A successful identity proof should use clear mathematical reasoning.

Consider:

(sec θ − cos θ)/tan θ = sin θ

We will verify this identity.

Start with the Left-Hand Side

(sec θ − cos θ)/tan θ

Replace Secant and Tangent

= (1/cos θ − cos θ)/(sin θ/cos θ)

Combine the Numerator

1/cos θ − cos θ

= (1 − cos² θ)/cos θ

Therefore:

= [(1 − cos² θ)/cos θ]/(sin θ/cos θ)

Apply the Pythagorean Identity

Since:

1 − cos² θ = sin² θ

we obtain:

= (sin² θ/cos θ)/(sin θ/cos θ)

Simplify

= (sin² θ/cos θ) × (cos θ/sin θ)

Cancel common factors:

= sin θ

The identity is verified wherever the original left-hand side is defined, requiring sin θ ≠ 0 and cos θ ≠ 0.

Choosing the Best Identity

Different expressions suggest different approaches.

The best method is usually the one that reduces the number of different trigonometric functions or simplifies the algebra.

Using Identities to Determine Unknown Values

Trigonometric identities can help us calculate unknown function values.

Worked Example

Suppose:

cos θ = −12/13

and θ lies in Quadrant II.

Find sin θ, tan θ, and sec θ.

Find Sine

Use:

sin² θ + cos² θ = 1

Substitute:

sin² θ + (−12/13)² = 1

sin² θ + 144/169 = 1

sin² θ = 25/169

sin θ = ±5/13

Since θ lies in Quadrant II, sine is positive.

sin θ = 5/13

Find Tangent

tan θ = sin θ/cos θ

= (5/13)/(−12/13)

tan θ = −5/12

Find Secant

sec θ = 1/cos θ

= 1/(−12/13)

sec θ = −13/12

The signs agree with Quadrant II.

Identities and the Unit Circle

The unit circle provides a geometric interpretation of the fundamental identities.

For any angle θ:

x = cos θ

y = sin θ

Since every point on the unit circle satisfies:

x² + y² = 1

we obtain:

sin² θ + cos² θ = 1

The quotient identities follow from the ratio of coordinates:

tan θ = y/x

cot θ = x/y

The reciprocal identities follow from taking the reciprocals of the coordinate-based definitions.

The unit circle therefore connects all three groups of identities through a single geometric model.

Applications in Physics and Engineering

Trigonometric identities are useful whenever calculations involve angles, vectors, waves, or circular motion.

Resolving Forces

A force F acting at an angle θ can be separated into components:

Fₓ = F cos θ

Fᵧ = F sin θ

Using the Pythagorean identity:

Fₓ² + Fᵧ²

= F²cos² θ + F²sin² θ

Factor:

= F²(cos² θ + sin² θ)

Since:

cos² θ + sin² θ = 1

we obtain:

Fₓ² + Fᵧ² = F²

This shows that the magnitude of the force is consistent with the Pythagorean theorem.

Circular Motion

The position of an object moving in a circle of radius r can be written:

x = r cos θ

y = r sin θ

Squaring and adding:

x² + y² = r²(cos² θ + sin² θ)

Therefore:

x² + y² = r²

This confirms that the object remains at a constant distance r from the centre.

Waves and Oscillations

Sine and cosine functions are used to describe oscillations.

For example:

y(t) = A sin(ωt)

The corresponding cosine function is related through a phase shift.

Trigonometric identities help simplify expressions involving wave motion, energy, and oscillatory systems.

Checking Identities Numerically

Substituting an angle into both sides of an identity can help check whether a proposed relationship is plausible.

For example:

1 + tan² θ = sec² θ

At θ = 45°:

Left-hand side:

1 + tan² 45°

= 1 + 1²

= 2

Right-hand side:

sec² 45°

= (√2)²

= 2

Both sides agree.

However, checking one or several angles does not prove an identity.

A mathematical proof must establish that the relationship is true for every angle in the appropriate domain.

Numerical checks are useful for detecting errors, but algebraic reasoning is required for verification.

A Strategy for Verifying Trigonometric Identities

When verifying a trigonometric identity:

1. Examine both sides of the equation.

2. Identify the more complicated side.

3. Begin simplifying that side.

4. Apply reciprocal, quotient, or Pythagorean identities.

5. Rewrite expressions using sine and cosine when useful.

6. Apply algebraic techniques such as factoring or common denominators.

7. Simplify carefully without dividing by expressions that may be zero.

8. Continue until the expression matches the other side.

9. State any necessary domain restrictions.

10. Conclude that the identity has been verified.

Important Principle

It is generally clearer to transform one side into the other rather than manipulate both sides independently.

This makes the logical steps of the proof easier to follow.

Common Mistakes

Confusing identities with equations

An identity is true for all values in its domain. An equation may only be true for specific values.

Forgetting reciprocal relationships

Cosecant is the reciprocal of sine, not cosine.

Using the wrong quotient identity

Tangent is sine divided by cosine. Cotangent is cosine divided by sine.

Incorrectly applying the Pythagorean identity

The correct identity is:

sin² θ + cos² θ = 1

not:

sin θ + cos θ = 1

Forgetting to square the entire trigonometric value

sin² θ means (sin θ)².

Assuming sin² θ = sin(θ²)

These expressions are not equivalent.

Cancelling terms incorrectly

Cancellation applies to common factors, not individual terms separated by addition or subtraction.

Dividing by zero

Always consider where denominators may equal zero.

Forgetting quadrant signs

Taking a square root may produce positive and negative possibilities.

Using numerical examples as proof

Checking a few angles does not establish an identity.

Changing both sides without clear justification

It is usually easier to simplify one side until it matches the other.

Failing to factor expressions

Factoring can reveal Pythagorean identities and common factors.

Key Terms

Trigonometric Identity: An equation involving trigonometric functions that is true for every value where both sides are defined.

Reciprocal Identity: An identity connecting a trigonometric function to its reciprocal.

Quotient Identity: An identity expressing tangent or cotangent as a ratio of sine and cosine.

Pythagorean Identity: An identity derived from the relationship sin² θ + cos² θ = 1.

Sine: A trigonometric function representing the vertical coordinate on the unit circle.

Cosine: A trigonometric function representing the horizontal coordinate on the unit circle.

Tangent: The ratio of sine to cosine.

Cosecant: The reciprocal of sine.

Secant: The reciprocal of cosine.

Cotangent: The reciprocal of tangent.

Unit Circle: A circle of radius 1 centred at the origin.

Pythagorean Theorem: The relationship a² + b² = c² in a right triangle.

Equivalent Expressions: Expressions that have the same value wherever both are defined.

Simplification: Rewriting an expression in a more useful or less complicated form.

Verification: Demonstrating that a proposed identity is mathematically valid.

Mathematical Proof: A logical sequence of steps establishing that a statement is true.

Factorisation: Rewriting an expression as a product of factors.

Common Denominator: A shared denominator used when combining fractions.

Difference of Squares: The algebraic identity a² − b² = (a − b)(a + b).

Conjugate: An expression formed by changing the sign between two terms in a binomial.

Domain Restriction: A condition excluding input values for which an expression is undefined.

Quadrant: One of the four regions of the Cartesian coordinate plane.

Reference Angle: The acute angle between an angle's terminal side and the x-axis.

Trigonometric Expression: An expression containing one or more trigonometric functions.

Vector Component: The portion of a vector acting along a specified direction.

Periodic Function: A function whose values repeat at regular intervals.

Check Your Understanding

1. Define a trigonometric identity and explain how it differs from a trigonometric equation.

2. Write the three fundamental reciprocal identities.

3. Write the two quotient identities.

4. State the three Pythagorean identities.

5. If sin θ = 4/5, determine csc θ.

6. If cos θ = −3/7, determine sec θ.

7. Simplify:

sin θ × cot θ

8. Simplify:

tan θ/sec θ

9. Use a Pythagorean identity to simplify:

1 − cos² θ

10. Simplify:

sec² θ − tan² θ

11. Verify:

(1 − sin² θ)/cos θ = cos θ

State the domain restriction.

12. Verify:

tan θ × cos θ = sin θ

State the domain restriction.

13. Simplify:

(1 − sin θ)(1 + sin θ)

14. Given cos θ = −5/13 and θ lies in Quadrant II, determine sin θ, tan θ, and csc θ.

15. Verify the identity:

(sec θ − cos θ)/tan θ = sin θ

Show each step clearly and identify where the original expression is defined.

Key Takeaways

• Trigonometric identities are equations that are true for all angles where both sides are defined.

• The three main groups of fundamental identities are reciprocal, quotient, and Pythagorean identities.

• Reciprocal identities connect sine with cosecant, cosine with secant, and tangent with cotangent.

• The quotient identity for tangent is tan θ = sin θ/cos θ.

• The quotient identity for cotangent is cot θ = cos θ/sin θ.

• The fundamental Pythagorean identity is sin² θ + cos² θ = 1.

• The second Pythagorean identity is 1 + tan² θ = sec² θ.

• The third Pythagorean identity is 1 + cot² θ = csc² θ.

• Pythagorean identities can be rearranged to calculate unknown trigonometric values.

• Quadrant information is needed to determine the correct signs of values obtained by square roots.

• Trigonometric identities can simplify complicated expressions.

• Rewriting expressions using sine and cosine is often an effective simplification strategy.

• Algebraic techniques such as factoring and common denominators are important in identity proofs.

• A trigonometric identity can be verified by transforming one side into the other.

• Numerical substitution can help check an identity but does not constitute proof.

• Domain restrictions must be respected when dividing or cancelling expressions.

• The unit circle provides the geometric foundation for the fundamental trigonometric identities.

• Trigonometric identities are useful in vector calculations, circular motion, waves, and oscillations.

• Mastering fundamental trigonometric identities is essential for solving advanced trigonometric equations and developing the algebraic skills required in calculus.

2. Sum and Difference Identities

Sum and Difference Identities

Trigonometric sum and difference identities allow us to calculate the sine, cosine, and tangent of angles formed by adding or subtracting two other angles.

For example, we may know the exact values of sin 30°, cos 45°, and tan 60°, but what about sin 75° or cos 15°?

These angles are not among the most commonly memorised special angles. However, they can be expressed as combinations of familiar angles:

75° = 45° + 30°

15° = 45° − 30°

Using sum and difference identities, we can calculate their exact trigonometric values without a calculator.

These identities are important in advanced trigonometry, calculus, geometry, physics, engineering, navigation, and the study of waves.

Understanding Compound Angles

A compound angle is an angle formed by adding or subtracting two angles.

Examples include:

• 75° = 45° + 30°

• 15° = 45° − 30°

• 105° = 60° + 45°

• 165° = 120° + 45°

• 7π/12 = π/3 + π/4

The goal is to express an unfamiliar angle as a combination of angles with known trigonometric values.

The most useful special angles are:

0°, 30°, 45°, 60°, and 90°.

Their exact values are:

By combining these values, we can evaluate many additional angles exactly.

Sine of a Sum of Angles

The sine addition identity is:

sin(A + B) = sin A cos B + cos A sin B

This identity allows us to calculate the sine of a sum using the sine and cosine of the individual angles.

Notice that the right-hand side contains two products added together.

Worked Example: Finding sin 75°

Express 75° as:

75° = 45° + 30°

Use the identity:

sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°

Substitute the exact values:

= (√2/2)(√3/2) + (√2/2)(1/2)

Multiply:

= √6/4 + √2/4

Combine:

sin 75° = (√6 + √2)/4

This is an exact value.

Using a calculator:

sin 75° ≈ 0.9659

The exact expression gives the same result without rounding.

Why This Identity Matters

The identity shows that the sine of a compound angle depends on both the sine and cosine of its component angles.

It is not generally true that:

sin(A + B) = sin A + sin B

For example:

sin(30° + 60°) = sin 90° = 1

But:

sin 30° + sin 60° = 1/2 + √3/2

which is not equal to 1.

Sine of a Difference of Angles

The sine subtraction identity is:

sin(A − B) = sin A cos B − cos A sin B

This is similar to the addition identity, except the two products are subtracted.

Worked Example: Finding sin 15°

Express:

15° = 45° − 30°

Use:

sin(45° − 30°) = sin 45° cos 30° − cos 45° sin 30°

Substitute:

= (√2/2)(√3/2) − (√2/2)(1/2)

Multiply:

= √6/4 − √2/4

Therefore:

sin 15° = (√6 − √2)/4

Approximately:

sin 15° ≈ 0.2588

Comparing the Results

Notice the relationship:

sin 75° = (√6 + √2)/4

sin 15° = (√6 − √2)/4

The difference in signs comes directly from the addition and subtraction identities.

Cosine of a Sum of Angles

The cosine addition identity is:

cos(A + B) = cos A cos B − sin A sin B

An important difference from the sine identity is that cosine addition involves subtraction.

Worked Example: Finding cos 75°

Express:

75° = 45° + 30°

Use:

cos(45° + 30°) = cos 45° cos 30° − sin 45° sin 30°

Substitute:

= (√2/2)(√3/2) − (√2/2)(1/2)

Simplify:

= √6/4 − √2/4

Therefore:

cos 75° = (√6 − √2)/4

Approximately:

cos 75° ≈ 0.2588

Connection to Complementary Angles

Notice:

cos 75° = sin 15°

This occurs because:

75° + 15° = 90°

For complementary angles:

sin θ = cos(90° − θ)

and:

cos θ = sin(90° − θ)

These relationships are useful for checking exact values.

Cosine of a Difference of Angles

The cosine subtraction identity is:

cos(A − B) = cos A cos B + sin A sin B

Notice that cosine subtraction involves addition.

This sign reversal is one of the most important features to remember.

Worked Example: Finding cos 15°

Express:

15° = 45° − 30°

Use:

cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30°

Substitute:

= (√2/2)(√3/2) + (√2/2)(1/2)

Simplify:

= √6/4 + √2/4

Therefore:

cos 15° = (√6 + √2)/4

Approximately:

cos 15° ≈ 0.9659

Summary of Sine and Cosine Identities

A useful memory rule is:

• Sine keeps the sign: Addition uses +, subtraction uses −.

• Cosine changes the sign: Addition uses −, subtraction uses +.

Tangent of a Sum of Angles

The tangent addition identity is:

tan(A + B) = (tan A + tan B)/(1 − tan A tan B)

This formula can be derived from the sine and cosine addition identities.

Because:

tan(A + B) = sin(A + B)/cos(A + B)

we substitute:

= (sin A cos B + cos A sin B)/(cos A cos B − sin A sin B)

Divide the numerator and denominator by cos A cos B:

= (tan A + tan B)/(1 − tan A tan B)

Therefore:

tan(A + B) = (tan A + tan B)/(1 − tan A tan B)

This form requires tan A and tan B to be defined and the denominator to be nonzero.

Worked Example: Finding tan 75°

Express:

75° = 45° + 30°

Use:

tan(45° + 30°) = (tan 45° + tan 30°)/(1 − tan 45° tan 30°)

Substitute:

= (1 + √3/3)/(1 − √3/3)

Multiply numerator and denominator by 3:

= (3 + √3)/(3 − √3)

Rationalise using the conjugate:

= [(3 + √3)(3 + √3)]/[(3 − √3)(3 + √3)]

Expand:

= (9 + 6√3 + 3)/(9 − 3)

= (12 + 6√3)/6

Therefore:

tan 75° = 2 + √3

Approximately:

tan 75° ≈ 3.732

Tangent of a Difference of Angles

The tangent subtraction identity is:

tan(A − B) = (tan A − tan B)/(1 + tan A tan B)

Notice that the denominator contains addition.

Worked Example: Finding tan 15°

Express:

15° = 45° − 30°

Use:

tan(45° − 30°) = (tan 45° − tan 30°)/(1 + tan 45° tan 30°)

Substitute:

= (1 − √3/3)/(1 + √3/3)

Multiply numerator and denominator by 3:

= (3 − √3)/(3 + √3)

Rationalise:

= [(3 − √3)(3 − √3)]/[(3 + √3)(3 − √3)]

Expand:

= (9 − 6√3 + 3)/(9 − 3)

= (12 − 6√3)/6

Therefore:

tan 15° = 2 − √3

Approximately:

tan 15° ≈ 0.2679

Tangent Identity Summary

The signs in the numerator and denominator are opposite.

Deriving the Identities Using Rotations

Sum and difference identities can be understood geometrically through rotations in the coordinate plane.

A point on the unit circle at angle A has coordinates:

(cos A, sin A)

Rotating this point counterclockwise by another angle B produces the point:

(cos(A + B), sin(A + B))

A rotation by angle B transforms coordinates according to:

x′ = x cos B − y sin B

y′ = x sin B + y cos B

Substitute:

x = cos A

y = sin A

Then:

x′ = cos A cos B − sin A sin B

y′ = cos A sin B + sin A cos B

But the rotated point has coordinates:

x′ = cos(A + B)

y′ = sin(A + B)

Therefore:

cos(A + B) = cos A cos B − sin A sin B

sin(A + B) = sin A cos B + cos A sin B

The subtraction identities follow by replacing B with −B and using:

cos(−B) = cos B

sin(−B) = −sin B

This derivation shows that the identities describe how horizontal and vertical coordinates change when rotations are combined.

Exact Values for Angles Beyond the First Quadrant

Sum and difference identities are also useful for angles in Quadrants II, III, and IV.

Worked Example: Finding sin 105°

Express:

105° = 60° + 45°

Use:

sin(60° + 45°) = sin 60° cos 45° + cos 60° sin 45°

Substitute:

= (√3/2)(√2/2) + (1/2)(√2/2)

Therefore:

sin 105° = (√6 + √2)/4

The answer is positive, as expected in Quadrant II.

Worked Example: Finding cos 105°

Use:

cos(60° + 45°) = cos 60° cos 45° − sin 60° sin 45°

Substitute:

= (1/2)(√2/2) − (√3/2)(√2/2)

Therefore:

cos 105° = (√2 − √6)/4

This value is negative, which is consistent with Quadrant II.

Worked Example: Finding tan 165°

Express:

165° = 120° + 45°

Since:

tan 120° = −√3

tan 45° = 1

Use:

tan 165° = (−√3 + 1)/(1 − (−√3)(1))

= (1 − √3)/(1 + √3)

Rationalise:

= [(1 − √3)²]/(1 − 3)

= (4 − 2√3)/(−2)

Therefore:

tan 165° = √3 − 2

The answer is negative, as expected in Quadrant II.

Exact Values in Radians

Sum and difference identities work equally well with radians.

For example:

5π/12 = π/6 + π/4

This corresponds to:

75° = 30° + 45°

Worked Example

Evaluate:

sin(5π/12)

Use:

sin(π/6 + π/4)

= sin(π/6)cos(π/4) + cos(π/6)sin(π/4)

Substitute:

= (1/2)(√2/2) + (√3/2)(√2/2)

Therefore:

sin(5π/12) = (√2 + √6)/4

Worked Example

Evaluate:

cos(π/12)

Since:

π/12 = π/4 − π/6

Use:

cos(π/4 − π/6)

= cos(π/4)cos(π/6) + sin(π/4)sin(π/6)

Substitute:

= (√2/2)(√3/2) + (√2/2)(1/2)

Therefore:

cos(π/12) = (√6 + √2)/4

Simplifying Trigonometric Expressions

Sum and difference identities can be used in reverse to simplify expressions.

Worked Example

Simplify:

sin A cos B + cos A sin B

Recognise the sine addition identity.

Therefore:

sin A cos B + cos A sin B = sin(A + B)

Worked Example

Simplify:

cos A cos B + sin A sin B

Recognise the cosine subtraction identity.

Therefore:

cos A cos B + sin A sin B = cos(A − B)

Worked Example

Simplify:

cos x cos 30° − sin x sin 30°

Use the cosine addition identity:

= cos(x + 30°)

This is much shorter than the original expression.

Verifying Identities Using Compound Angles

Sum and difference identities can be combined with other trigonometric identities.

Worked Example

Verify:

sin(x + 90°) = cos x

Use the sine addition identity:

sin(x + 90°)

= sin x cos 90° + cos x sin 90°

Since:

cos 90° = 0

sin 90° = 1

we obtain:

= sin x(0) + cos x(1)

= cos x

The identity is verified.

Worked Example

Verify:

cos(x + 180°) = −cos x

Use:

cos(x + 180°)

= cos x cos 180° − sin x sin 180°

Since:

cos 180° = −1

sin 180° = 0

we obtain:

= cos x(−1) − sin x(0)

= −cos x

The identity is verified.

Worked Example

Verify:

sin(x − 90°) = −cos x

Use:

sin(x − 90°)

= sin x cos 90° − cos x sin 90°

= sin x(0) − cos x(1)

= −cos x

Complementary and Supplementary Angle Identities

Sum and difference identities can be used to derive relationships between related angles.

Complementary Angles

Two angles are complementary if they add to 90°.

Using the sine difference identity:

sin(90° − θ)

= sin 90° cos θ − cos 90° sin θ

= cos θ

Therefore:

sin(90° − θ) = cos θ

Similarly:

cos(90° − θ) = sin θ

These are called cofunction identities.

Supplementary Angles

Two angles are supplementary if they add to 180°.

Using the sine difference identity:

sin(180° − θ)

= sin 180° cos θ − cos 180° sin θ

= 0 − (−1)sin θ

Therefore:

sin(180° − θ) = sin θ

Similarly:

cos(180° − θ) = −cos θ

These relationships help evaluate trigonometric functions in different quadrants.

Finding Unknown Trigonometric Values

Sum and difference identities can be applied when the individual angles are not special angles.

Suppose:

sin A = 3/5

cos A = 4/5

sin B = 5/13

cos B = 12/13

Assume A and B are acute angles.

Find sin(A + B).

Use:

sin(A + B) = sin A cos B + cos A sin B

Substitute:

= (3/5)(12/13) + (4/5)(5/13)

= 36/65 + 20/65

Therefore:

sin(A + B) = 56/65

Finding Cosine

Find cos(A + B).

Use:

cos(A + B) = cos A cos B − sin A sin B

Substitute:

= (4/5)(12/13) − (3/5)(5/13)

= 48/65 − 15/65

Therefore:

cos(A + B) = 33/65

Finding Tangent

Use:

tan(A + B) = sin(A + B)/cos(A + B)

= (56/65)/(33/65)

Therefore:

tan(A + B) = 56/33

This example shows how compound-angle identities can be applied without knowing the actual angle measurements.

Compound Angles and Quadrant Signs

When working with compound angles, the quadrant of the resulting angle is important.

For example:

A = 120°

B = 45°

Then:

A + B = 165°

This lies in Quadrant II.

Therefore:

• Sine is positive.

• Cosine is negative.

• Tangent is negative.

The signs provide a useful check on exact calculations.

Worked Example

Find cos 165°.

Express:

165° = 120° + 45°

Use:

cos(120° + 45°)

= cos 120° cos 45° − sin 120° sin 45°

Substitute:

= (−1/2)(√2/2) − (√3/2)(√2/2)

Therefore:

cos 165° = −(√2 + √6)/4

The result is negative, consistent with Quadrant II.

Applications in Geometry

Sum and difference identities can help solve geometric problems involving angles formed by combining or separating rotations.

Worked Example: Finding an Exact Direction Component

A displacement vector has magnitude 20 m and makes an angle of 75° with the positive horizontal direction.

Find its exact horizontal component.

The horizontal component is:

x = 20 cos 75°

Using:

cos 75° = (√6 − √2)/4

Substitute:

x = 20(√6 − √2)/4

Therefore:

x = 5(√6 − √2) m

Approximately:

x ≈ 5.18 m

The vertical component is:

y = 20 sin 75°

Using:

sin 75° = (√6 + √2)/4

Therefore:

y = 5(√6 + √2) m

Approximately:

y ≈ 19.32 m

Applications in Physics: Combining Oscillations

Trigonometric identities are useful when analysing waves and oscillations.

Consider two signals:

y₁ = sin(ωt)

y₂ = cos(ωt)

A phase-shifted signal can be expressed as:

y = sin(ωt + φ)

Using the sine addition identity:

sin(ωt + φ)

= sin(ωt)cos φ + cos(ωt)sin φ

This shows that a phase-shifted sine wave can be written as a combination of sine and cosine waves.

Example

Suppose:

φ = 30°

Then:

sin(ωt + 30°)

= sin(ωt)cos 30° + cos(ωt)sin 30°

Substitute:

= (√3/2)sin(ωt) + (1/2)cos(ωt)

Therefore:

sin(ωt + 30°) = (√3/2)sin(ωt) + (1/2)cos(ωt)

This relationship is important in wave interference, alternating current, and signal processing.

Applications in Engineering: Rotations

In engineering and computer graphics, points and vectors are often rotated through specified angles.

A point (x, y) rotated counterclockwise by angle θ has new coordinates:

x′ = x cos θ − y sin θ

y′ = x sin θ + y cos θ

These formulas are closely related to the sum identities.

Worked Example

Rotate the point:

(3, 4)

counterclockwise by 30°.

Use:

x′ = 3 cos 30° − 4 sin 30°

= 3(√3/2) − 4(1/2)

Therefore:

x′ = (3√3 − 4)/2

For the vertical coordinate:

y′ = 3 sin 30° + 4 cos 30°

= 3(1/2) + 4(√3/2)

Therefore:

y′ = (3 + 4√3)/2

The rotated point is:

((3√3 − 4)/2, (3 + 4√3)/2)

These transformations are used in robotics, animation, navigation, and mechanical design.

Solving Equations with Compound Angles

Sum and difference identities can also be used to solve trigonometric equations.

Worked Example

Solve:

sin(x + 30°) = 1/2

for:

0° ≤ x < 360°

Let:

u = x + 30°

Then:

sin u = 1/2

The general solutions are:

u = 30° + 360°n

or:

u = 150° + 360°n

Substitute back.

First family:

x + 30° = 30° + 360°n

x = 360°n

Second family:

x + 30° = 150° + 360°n

x = 120° + 360°n

Within the given interval:

x = 0° or 120°

Worked Example

Solve:

cos(x − 45°) = √2/2

for:

0° ≤ x < 360°

Let:

u = x − 45°

Then:

cos u = √2/2

The general solutions are:

u = 45° + 360°n

or:

u = −45° + 360°n

Substitute back.

First family:

x − 45° = 45° + 360°n

x = 90° + 360°n

Second family:

x − 45° = −45° + 360°n

x = 360°n

Therefore:

x = 0° or 90°

A Strategy for Using Sum and Difference Identities

When solving a compound-angle problem:

1. Identify the trigonometric function involved.

2. Determine whether the angle is a sum or difference.

3. Express unfamiliar angles using known special angles when possible.

4. Select the correct identity.

5. Substitute exact trigonometric values.

6. Simplify products and fractions carefully.

7. Rationalise denominators when appropriate.

8. Check the sign using the quadrant.

9. Use a calculator only to verify the result if an exact value is required.

10. State the final answer clearly.

Example

Evaluate:

sin 165°

Express:

165° = 120° + 45°

Apply:

sin(120° + 45°)

= sin 120° cos 45° + cos 120° sin 45°

Substitute:

= (√3/2)(√2/2) + (−1/2)(√2/2)

Simplify:

= √6/4 − √2/4

Therefore:

sin 165° = (√6 − √2)/4

The answer is positive, as expected in Quadrant II.

Common Mistakes

Assuming trigonometric functions distribute over addition

In general:

sin(A + B) ≠ sin A + sin B

and:

cos(A + B) ≠ cos A + cos B

Confusing sine and cosine identities

Sine identities use mixed products of sine and cosine. Cosine identities use cosine–cosine and sine–sine products.

Using the wrong signs

Cosine addition uses subtraction. Cosine subtraction uses addition.

Forgetting the tangent denominator

The tangent identities involve both a numerator and denominator.

Using the same sign in both parts of the tangent identity

The numerator and denominator use opposite signs.

Using incorrect exact values

Review the special-angle values before substituting.

Forgetting quadrant signs

Angles in Quadrants II, III, and IV may have negative trigonometric values.

Rounding too early

Keep radicals and fractions until the final answer.

Confusing degrees and radians

Ensure all angles use consistent units.

Assuming every compound angle has a convenient exact form

Some angles do not simplify neatly using familiar special-angle combinations.

Ignoring undefined tangent values

The tangent sum and difference formulas require the individual tangent values and the resulting denominator to be defined.

Making algebraic errors when rationalising

Multiply both numerator and denominator by the conjugate.

Key Terms

Compound Angle: An angle formed by adding or subtracting two angles.

Angle Sum Identity: A formula expressing a trigonometric function of A + B using functions of A and B.

Angle Difference Identity: A formula expressing a trigonometric function of A − B using functions of A and B.

Sine Addition Identity: The relationship sin(A + B) = sin A cos B + cos A sin B.

Sine Subtraction Identity: The relationship sin(A − B) = sin A cos B − cos A sin B.

Cosine Addition Identity: The relationship cos(A + B) = cos A cos B − sin A sin B.

Cosine Subtraction Identity: The relationship cos(A − B) = cos A cos B + sin A sin B.

Tangent Addition Identity: The relationship tan(A + B) = (tan A + tan B)/(1 − tan A tan B), where defined.

Tangent Subtraction Identity: The relationship tan(A − B) = (tan A − tan B)/(1 + tan A tan B), where defined.

Exact Trigonometric Value: A trigonometric value expressed without rounding, often using fractions and radicals.

Special Angle: An angle with commonly known exact trigonometric values, such as 30°, 45°, or 60°.

Unit Circle: A circle of radius 1 centred at the origin.

Reference Angle: The acute angle formed between an angle's terminal side and the x-axis.

Quadrant: One of the four regions of the Cartesian coordinate plane.

Trigonometric Identity: An equation involving trigonometric functions that is true wherever both sides are defined.

Equivalent Expressions: Expressions that produce the same values over their common domain.

Rationalising a Denominator: Removing a radical from the denominator of a fraction.

Conjugate: A binomial formed by changing the sign between two terms.

Complementary Angles: Two angles whose sum is 90°.

Supplementary Angles: Two angles whose sum is 180°.

Cofunction Identity: A relationship between trigonometric functions of complementary angles.

Phase Shift: A horizontal displacement of a periodic function.

Rotation Transformation: A transformation that turns a point or vector around a fixed centre.

Vector Component: The horizontal or vertical part of a vector.

Periodic Function: A function whose values repeat at regular intervals.

Sinusoidal Function: A function represented using sine or cosine.

Angular Frequency: The rate of angular change, commonly represented by ω.

Check Your Understanding

1. Write the sine addition and sine subtraction identities.

2. Write the cosine addition and cosine subtraction identities.

3. Write the tangent addition and tangent subtraction identities.

4. Explain why sin(A + B) is not generally equal to sin A + sin B.

5. Use a sum identity to determine the exact value of sin 75°.

6. Use a difference identity to determine the exact value of cos 15°.

7. Determine the exact value of tan 75°.

8. Determine the exact value of tan 15°.

9. Find the exact value of sin 105°.

10. Find the exact value of cos 165°.

11. Simplify:

sin x cos 30° + cos x sin 30°

12. Verify:

sin(x + 90°) = cos x

13. Given sin A = 3/5, cos A = 4/5, sin B = 5/13, and cos B = 12/13, find sin(A + B) and cos(A − B).

14. Solve:

sin(x + 45°) = √2/2

for 0° ≤ x < 360°.

15. A vector of magnitude 40 N acts at an angle of 75° above the horizontal. Use sum and difference identities to determine its exact horizontal and vertical components.

Key Takeaways

• Sum and difference identities allow trigonometric functions of compound angles to be expressed using the functions of individual angles.

• A compound angle is formed by adding or subtracting two angles.

• The sine addition identity is sin(A + B) = sin A cos B + cos A sin B.

• The sine subtraction identity is sin(A − B) = sin A cos B − cos A sin B.

• The cosine addition identity is cos(A + B) = cos A cos B − sin A sin B.

• The cosine subtraction identity is cos(A − B) = cos A cos B + sin A sin B.

• The tangent addition identity is tan(A + B) = (tan A + tan B)/(1 − tan A tan B), where defined.

• The tangent subtraction identity is tan(A − B) = (tan A − tan B)/(1 + tan A tan B), where defined.

• Sine identities retain the addition or subtraction sign between their two products.

• Cosine identities use the opposite sign between their two products.

• Tangent identities use opposite signs in their numerators and denominators.

• Exact values for angles such as 15°, 75°, and 105° can be derived from familiar special angles.

• Compound-angle identities work with both degrees and radians.

• The identities can be used in reverse to simplify trigonometric expressions.

• The unit circle and coordinate rotations provide geometric explanations for the identities.

• Complementary and supplementary angle identities can be derived from sum and difference formulas.

• Quadrant signs provide a useful check on calculated values.

• Compound-angle identities are useful for solving trigonometric equations.

• They are also used to analyse vectors, rotations, waves, oscillations, and phase shifts.

• Mastering sum and difference identities prepares students for double-angle identities, advanced trigonometric equations, and calculus.

3. Double-Angle and Half-Angle Identities

4. Solving Trigonometric Equations

Solving Trigonometric Equations

Trigonometric equations are equations that contain one or more trigonometric functions, such as sine, cosine, or tangent, with an unknown angle.

For example:

sin θ = 1/2

Unlike many algebraic equations, trigonometric equations often have multiple solutions because trigonometric functions are periodic. Their values repeat at regular intervals.

For example:

sin 30° = 1/2

sin 150° = 1/2

sin 390° = 1/2

All three angles satisfy the same equation.

Solving trigonometric equations therefore requires more than finding a single angle. We must understand the unit circle, use inverse trigonometric functions, identify solutions in different quadrants, and account for periodicity.

These skills are essential in mathematics, physics, engineering, navigation, and the study of waves and oscillations.

Understanding Trigonometric Equations

A trigonometric equation contains a trigonometric function of an unknown angle.

Examples include:

• sin x = 1/2

• cos x = −√2/2

• tan x = √3

• 2sin x − 1 = 0

• sin² x + sin x = 0

• cos 2x = 1/2

The goal is to determine the values of x that make the equation true.

A trigonometric equation may have:

• No solutions.

• One solution within a specified interval.

• Several solutions within a specified interval.

• Infinitely many solutions when no interval is specified.

Equations Compared with Identities

A trigonometric equation is true only for particular values of x.

For example:

sin x = 1/2

A trigonometric identity is true for every value where both sides are defined.

For example:

sin² x + cos² x = 1

Identities are often used to simplify equations before solving them.

Why Trigonometric Equations Have Multiple Solutions

Trigonometric functions are periodic.

The sine and cosine functions repeat every 360°, or 2π radians.

The tangent function repeats every 180°, or π radians.

Therefore:

sin(x + 360°) = sin x

cos(x + 360°) = cos x

tan(x + 180°) = tan x

For example:

sin 30° = sin 390° = sin 750° = 1/2

These angles differ by complete revolutions.

General Solutions

When no interval is specified, solutions are often written using an integer n.

For example:

x = 30° + 360°n

where n is any integer.

This represents infinitely many coterminal angles.

However, sine also equals 1/2 at 150°.

Therefore, the complete general solution is:

x = 30° + 360°n

or:

x = 150° + 360°n

where n is any integer.

Solving Basic Sine Equations

Consider:

sin x = 1/2

for:

0° ≤ x < 360°

Find the Reference Angle

Use inverse sine:

x = sin⁻¹(1/2)

The principal value is:

30°

Determine the Quadrants

Sine is positive in Quadrants I and II.

Therefore:

First solution:

x = 30°

Second solution:

x = 180° − 30°

x = 150°

Final Answer

x = 30°, 150°

Verifying the Solutions

sin 30° = 1/2

sin 150° = 1/2

Both values satisfy the original equation.

Solving Sine Equations with Negative Values

Consider:

sin x = −√3/2

for:

0° ≤ x < 360°

Find the Reference Angle

Since:

sin 60° = √3/2

the reference angle is:

60°

Determine the Quadrants

Sine is negative in Quadrants III and IV.

Therefore:

Quadrant III:

x = 180° + 60°

x = 240°

Quadrant IV:

x = 360° − 60°

x = 300°

Final Answer

x = 240°, 300°

Verification

sin 240° = −√3/2

sin 300° = −√3/2

Both solutions are correct.

Solving Basic Cosine Equations

Consider:

cos x = 1/2

for:

0° ≤ x < 360°

Find the Reference Angle

cos⁻¹(1/2) = 60°

Determine the Quadrants

Cosine is positive in Quadrants I and IV.

First solution:

x = 60°

Second solution:

x = 360° − 60°

x = 300°

Final Answer

x = 60°, 300°

Worked Example: Negative Cosine

Solve:

cos x = −√2/2

for:

0° ≤ x < 360°

The reference angle is 45°.

Cosine is negative in Quadrants II and III.

Therefore:

x = 180° − 45° = 135°

x = 180° + 45° = 225°

x = 135°, 225°

Solving Basic Tangent Equations

Consider:

tan x = 1

for:

0° ≤ x < 360°

Find the Reference Angle

tan⁻¹(1) = 45°

Determine the Quadrants

Tangent is positive in Quadrants I and III.

Therefore:

x = 45°

x = 180° + 45° = 225°

Final Answer

x = 45°, 225°

Tangent Periodicity

Tangent repeats every 180°.

Therefore, the general solution is:

x = 45° + 180°n

where n is any integer.

Worked Example: Negative Tangent

Solve:

tan x = −√3

for:

0° ≤ x < 360°

The reference angle is 60°.

Tangent is negative in Quadrants II and IV.

Therefore:

x = 180° − 60° = 120°

x = 360° − 60° = 300°

x = 120°, 300°

Using the Unit Circle to Find Solutions

The unit circle provides a useful way to determine which quadrants contain solutions.

For an angle x:

• cos x is the horizontal coordinate.

• sin x is the vertical coordinate.

• tan x = sin x/cos x, where cosine is nonzero.

The signs of the functions depend on the quadrant.

A useful memory rule is ASTC:

• Quadrant I: All positive.

• Quadrant II: Sine positive.

• Quadrant III: Tangent positive.

• Quadrant IV: Cosine positive.

Finding Solutions from Reference Angles

If the reference angle is α:

These formulas help locate solutions within one complete revolution.

Solving Equations in Radians

Trigonometric equations can also be solved using radians.

A complete revolution is:

2π radians

A half revolution is:

π radians

Worked Example

Solve:

sin x = √2/2

for:

0 ≤ x < 2π

The reference angle is:

π/4

Sine is positive in Quadrants I and II.

Therefore:

First solution:

x = π/4

Second solution:

x = π − π/4

x = 3π/4

Final Answer

x = π/4, 3π/4

Worked Example

Solve:

cos x = −1/2

for:

0 ≤ x < 2π

The reference angle is:

π/3

Cosine is negative in Quadrants II and III.

Therefore:

x = 2π/3

x = 4π/3

x = 2π/3, 4π/3

General Solutions for Sine, Cosine, and Tangent

When no interval is specified, we must include all possible solutions.

Let α be the principal angle returned by the appropriate inverse function.

Sine

If:

sin x = a

then:

x = α + 2πn

or:

x = π − α + 2πn

where:

α = arcsin(a)

and n is any integer.

This requires −1 ≤ a ≤ 1.

Cosine

If:

cos x = a

then:

x = ±α + 2πn

where:

α = arccos(a)

and n is any integer.

This requires −1 ≤ a ≤ 1.

Tangent

If:

tan x = a

then:

x = α + πn

where:

α = arctan(a)

and n is any integer.

General Solution Table

For sine and cosine, α is defined using their respective inverse functions.

These formulas can also be written in degrees by replacing 2π with 360° and π with 180°.

Solving Equations with Coefficients

Many trigonometric equations require algebraic rearrangement before finding the angle.

Worked Example

Solve:

2sin x − 1 = 0

for:

0° ≤ x < 360°

Isolate the Trigonometric Function

2sin x = 1

sin x = 1/2

Find the Solutions

The reference angle is 30°.

Sine is positive in Quadrants I and II.

Therefore:

x = 30°, 150°

Worked Example

Solve:

3cos x + 1 = 0

for:

0° ≤ x < 360°

Rearrange:

3cos x = −1

cos x = −1/3

Find the principal angle:

x = cos⁻¹(−1/3)

x ≈ 109.47°

Cosine is negative in Quadrants II and III.

The second solution is:

360° − 109.47° = 250.53°

Therefore:

x ≈ 109.47°, 250.53°

Equations with No Real Solutions

Not every trigonometric equation has a solution.

For example:

sin x = 2

Sine values always lie between −1 and 1.

Therefore, no real angle can satisfy this equation.

Similarly:

cos x = −1.5

has no real solutions.

Worked Example

Solve:

4sin x − 6 = 0

Rearrange:

4sin x = 6

sin x = 1.5

Since:

−1 ≤ sin x ≤ 1

the equation has:

No real solutions

Important Principle

Always check whether the required sine or cosine value is within the permitted range before calculating angles.

Tangent has no corresponding range restriction because tangent can take any real value.

Solving Equations Involving Multiple Angles

Some equations contain expressions such as:

sin 2x

cos 3x

tan(x − 30°)

These require careful consideration of the angle interval.

Worked Example

Solve:

sin 2x = 1/2

for:

0° ≤ x < 360°

Adjust the Interval

Since:

0° ≤ x < 360°

multiply the interval by 2:

0° ≤ 2x < 720°

Find All Solutions for 2x

Sine equals 1/2 at:

30°, 150°, 390°, 510°

within this interval.

Therefore:

2x = 30°, 150°, 390°, 510°

Divide each by 2:

x = 15°, 75°, 195°, 255°

Why Are There Four Solutions?

The function sin 2x completes two full cycles between 0° and 360°.

Therefore, there are more solutions than for sin x = 1/2 over the same interval.

Solving Cosine Equations with Multiple Angles

Worked Example

Solve:

cos 3x = 0

for:

0° ≤ x < 360°

Adjust the Interval

0° ≤ 3x < 1080°

Cosine equals zero at:

90°, 270°, 450°, 630°, 810°, 990°

Therefore:

3x = 90°, 270°, 450°, 630°, 810°, 990°

Divide by 3:

x = 30°, 90°, 150°, 210°, 270°, 330°

There are six solutions.

Solving Equations with Horizontal Translations

Some equations contain a phase shift.

Worked Example

Solve:

sin(x − 30°) = 1/2

for:

0° ≤ x < 360°

Let:

u = x − 30°

Then:

sin u = 1/2

The general solutions are:

u = 30° + 360°n

or:

u = 150° + 360°n

Substitute back:

x − 30° = 30° + 360°n

x = 60° + 360°n

or:

x − 30° = 150° + 360°n

x = 180° + 360°n

Within the interval:

x = 60°, 180°

Worked Example

Solve:

cos(x + 45°) = √2/2

for:

0° ≤ x < 360°

Let:

u = x + 45°

The general solutions are:

u = 45° + 360°n

or:

u = −45° + 360°n

Therefore:

x = 0° + 360°n

or:

x = −90° + 360°n

Within the interval:

x = 0°, 270°

Solving Equations Using Pythagorean Identities

Trigonometric identities can simplify equations containing several trigonometric functions.

Recall:

sin² x + cos² x = 1

Other useful forms include:

1 + tan² x = sec² x

1 + cot² x = csc² x

Worked Example

Solve:

sin² x + cos x = 1

for:

0° ≤ x < 360°

Use:

sin² x = 1 − cos² x

Substitute:

1 − cos² x + cos x = 1

Subtract 1:

−cos² x + cos x = 0

Factor:

cos x(1 − cos x) = 0

Therefore:

cos x = 0

or:

cos x = 1

First Case

cos x = 0

x = 90°, 270°

Second Case

cos x = 1

x = 0°

Final Answer

x = 0°, 90°, 270°

Solving Equations by Factoring

Factoring is a useful technique when an equation contains products or powers of trigonometric functions.

Worked Example

Solve:

sin² x − sin x = 0

for:

0° ≤ x < 360°

Factor:

sin x(sin x − 1) = 0

Therefore:

sin x = 0

or:

sin x − 1 = 0

First Case

sin x = 0

x = 0°, 180°

Second Case

sin x = 1

x = 90°

Final Answer

x = 0°, 90°, 180°

Important Warning

Do not divide both sides by sin x.

Doing so would remove the solutions where sin x = 0.

Factoring preserves all possible solutions.

Solving Quadratic Trigonometric Equations

Some trigonometric equations resemble quadratic equations.

For example:

2sin² x − 3sin x + 1 = 0

These can be solved using substitution or factoring.

Worked Example

Solve:

2sin² x − 3sin x + 1 = 0

for:

0° ≤ x < 360°

Treat Sine as a Variable

Let:

u = sin x

Then:

2u² − 3u + 1 = 0

Factor:

(2u − 1)(u − 1) = 0

Therefore:

u = 1/2

or:

u = 1

Substitute Back

sin x = 1/2

or:

sin x = 1

Find the Solutions

For sin x = 1/2:

x = 30°, 150°

For sin x = 1:

x = 90°

Final Answer

x = 30°, 90°, 150°

Solving Quadratic Equations Involving Cosine

Worked Example

Solve:

2cos² x + cos x − 1 = 0

for:

0° ≤ x < 360°

Let:

u = cos x

Then:

2u² + u − 1 = 0

Factor:

(2u − 1)(u + 1) = 0

Therefore:

u = 1/2

or:

u = −1

Substitute back:

cos x = 1/2

or:

cos x = −1

First Case

cos x = 1/2

x = 60°, 300°

Second Case

cos x = −1

x = 180°

Final Answer

x = 60°, 180°, 300°

Solving Equations Using Double-Angle Identities

Double-angle identities allow expressions involving 2x to be rewritten using functions of x.

Recall:

sin 2x = 2sin x cos x

cos 2x = 1 − 2sin² x

cos 2x = 2cos² x − 1

Worked Example

Solve:

sin 2x = sin x

for:

0° ≤ x < 360°

Use:

sin 2x = 2sin x cos x

Therefore:

2sin x cos x = sin x

Move everything to one side:

2sin x cos x − sin x = 0

Factor:

sin x(2cos x − 1) = 0

Therefore:

sin x = 0

or:

cos x = 1/2

First Case

sin x = 0

x = 0°, 180°

Second Case

cos x = 1/2

x = 60°, 300°

Final Answer

x = 0°, 60°, 180°, 300°

Solving Equations Using Cosine Double-Angle Identities

Worked Example

Solve:

cos 2x = cos x

for:

0° ≤ x < 360°

Use:

cos 2x = 2cos² x − 1

Substitute:

2cos² x − 1 = cos x

Rearrange:

2cos² x − cos x − 1 = 0

Factor:

(2cos x + 1)(cos x − 1) = 0

Therefore:

cos x = −1/2

or:

cos x = 1

First Case

cos x = −1/2

x = 120°, 240°

Second Case

cos x = 1

x = 0°

Final Answer

x = 0°, 120°, 240°

Solving Equations with Multiple Trigonometric Functions

Some equations contain sine and cosine together.

These may require factoring, division, or identities.

Worked Example

Solve:

sin x = cos x

for:

0° ≤ x < 360°

Where cos x ≠ 0, divide both sides by cos x:

sin x/cos x = 1

Therefore:

tan x = 1

The reference angle is 45°.

Tangent is positive in Quadrants I and III.

Therefore:

x = 45°, 225°

At angles where cos x = 0, sin x is ±1, so none of those excluded angles satisfy the original equation.

Worked Example

Solve:

√3 sin x = cos x

for:

0° ≤ x < 360°

Where cos x ≠ 0, divide by cos x:

√3 tan x = 1

Therefore:

tan x = 1/√3

tan x = √3/3

The reference angle is 30°.

Tangent is positive in Quadrants I and III.

Therefore:

x = 30°, 210°

Angles where cos x = 0 do not satisfy the original equation.

Verifying Trigonometric Solutions

Finding possible solutions is not always enough.

We should substitute them into the original equation to verify that they are valid.

Worked Example

Verify the solutions of:

2sin x − 1 = 0

We found:

x = 30°, 150°

Check x = 30°

2sin 30° − 1

= 2(1/2) − 1

= 1 − 1

= 0

The solution is valid.

Check x = 150°

2sin 150° − 1

= 2(1/2) − 1

= 0

The solution is valid.

Why Verification Matters

Verification can detect:

• Algebraic mistakes.

• Incorrect quadrant choices.

• Solutions outside the required interval.

• Invalid solutions introduced by squaring.

• Errors caused by dividing by zero.

Extraneous Solutions

An extraneous solution is a value obtained during algebraic manipulation that does not satisfy the original equation.

These may appear when squaring both sides of an equation.

Worked Example

Solve:

sin x = cos² x

for:

0° ≤ x < 360°

Use:

cos² x = 1 − sin² x

Therefore:

sin x = 1 − sin² x

Rearrange:

sin² x + sin x − 1 = 0

Let:

u = sin x

Then:

u² + u − 1 = 0

Use the quadratic formula:

u = (−1 ± √5)/2

The two values are:

u = (−1 + √5)/2 ≈ 0.6180

u = (−1 − √5)/2 ≈ −1.6180

Since sine must lie between −1 and 1, the second value is impossible.

Therefore:

sin x ≈ 0.6180

The reference angle is approximately:

38.17°

Sine is positive in Quadrants I and II.

Therefore:

x ≈ 38.17°, 141.83°

The rejected algebraic value is outside the range of sine and cannot produce a real solution.

Solving Equations Over Different Intervals

The interval determines which solutions must be included.

Consider:

sin x = 1/2

Interval: 0° ≤ x < 360°

Solutions:

30°, 150°

Interval: 0° ≤ x < 720°

Solutions:

30°, 150°, 390°, 510°

Interval: −180° ≤ x < 180°

Solutions:

−30°?

No. Since sin(−30°) = −1/2, it is not a solution.

The solutions in this interval are:

−210°?

This is outside the interval.

Instead, the correct solutions are:

30°, 150°

Interval: −360° ≤ x < 0°

Subtract 360° from the positive solutions:

30° − 360° = −330°

150° − 360° = −210°

Therefore:

x = −330°, −210°

Important Principle

Always check interval boundaries carefully.

If the interval uses:

0° ≤ x < 360°

then 0° is included, but 360° is excluded.

Solving Trigonometric Equations Graphically

Trigonometric equations can be interpreted as intersections of graphs.

For example:

sin x = 1/2

can be represented by:

y = sin x

and:

y = 1/2

The solutions are the x-coordinates where the two graphs intersect.

Graphical Method

To solve an equation graphically:

1. Graph the left-hand side.

2. Graph the right-hand side.

3. Identify the intersection points.

4. Read the corresponding x-values.

5. Check that the solutions lie within the required interval.

Advantages

Graphical methods are useful for:

• Estimating solutions.

• Checking algebraic answers.

• Understanding periodicity.

• Identifying the number of solutions.

• Solving equations that cannot easily be rearranged.

Limitations

Graphical solutions may be approximate.

For exact answers, algebraic methods and trigonometric identities are usually preferred.

Applications in Physics: Oscillatory Motion

Trigonometric equations are frequently used to determine when an oscillating object reaches a particular position.

Suppose a vibrating object has displacement:

x(t) = 5sin(πt/2)

where:

• x is displacement in centimetres.

• t is time in seconds.

Find when the displacement is 2.5 cm during the first 4 seconds.

Form the Equation

5sin(πt/2) = 2.5

Divide by 5:

sin(πt/2) = 1/2

Solve for the Angle

Let:

u = πt/2

Since 0 ≤ t ≤ 4:

0 ≤ u ≤ 2π

Sine equals 1/2 at:

u = π/6

and:

u = 5π/6

Solve for Time

First solution:

πt/2 = π/6

t = 1/3 s

Second solution:

πt/2 = 5π/6

t = 5/3 s

Final Answer

t = 1/3 s and 5/3 s

The object reaches a displacement of 2.5 cm twice during the first complete cycle.

Applications in Engineering: Rotating Machinery

A rotating component has angular position:

θ(t) = 120t

where θ is measured in degrees and t is measured in seconds.

A sensor activates whenever:

cos θ = 1/2

Find the activation times during the first 6 seconds.

Form the Equation

cos(120t) = 1/2

During the first 6 seconds:

0 ≤ t ≤ 6

Therefore:

0° ≤ 120t ≤ 720°

Find the Angles

Cosine equals 1/2 at:

60°, 300°, 420°, 660°

Solve for Time

120t = 60°

t = 0.5 s

120t = 300°

t = 2.5 s

120t = 420°

t = 3.5 s

120t = 660°

t = 5.5 s

Final Answer

t = 0.5 s, 2.5 s, 3.5 s, and 5.5 s

The sensor activates four times during the first 6 seconds.

Applications in Navigation and Geometry

Trigonometric equations are useful when determining directions and angles from known measurements.

Worked Example

A force has a horizontal component of 12 N and a magnitude of 20 N.

Find the possible directions relative to the positive horizontal axis for:

0° ≤ θ < 360°

Form the Equation

Fₓ = F cos θ

Substitute:

12 = 20cos θ

Divide:

cos θ = 12/20

cos θ = 0.6

Find the Reference Angle

θ = cos⁻¹(0.6)

θ ≈ 53.13°

Cosine is positive in Quadrants I and IV.

Therefore:

θ₁ ≈ 53.13°

θ₂ = 360° − 53.13°

θ₂ ≈ 306.87°

Final Answer

θ ≈ 53.13° or 306.87°

These represent two possible force directions with the same positive horizontal component.

A Strategy for Solving Trigonometric Equations

A systematic approach helps prevent missed or incorrect solutions.

Identify the Equation Type

Determine whether the equation involves:

• A single trigonometric function.

• An algebraic coefficient.

• A multiple angle.

• A phase shift.

• A trigonometric identity.

• A quadratic expression.

• Several trigonometric functions.

Simplify the Equation

Use algebra and identities to isolate or factor trigonometric expressions.

Find Reference Angles

Use exact special-angle values or inverse trigonometric functions.

Determine All Relevant Quadrants

Use the signs of sine, cosine, and tangent.

Account for Periodicity

Include all solutions in the specified interval.

Solve for the Original Variable

If the equation contains 2x, 3x, or x − 30°, remember to solve for x after finding the possible angle values.

Verify

Substitute solutions into the original equation.

State the Final Answer

List solutions in ascending order and include the appropriate units.

Common Mistakes

Finding only the principal value

Inverse trigonometric functions usually return one angle, but the equation may have several solutions.

Ignoring the specified interval

Only solutions within the given interval should be included.

Forgetting periodicity

Sine and cosine repeat every 360°, while tangent repeats every 180°.

Using incorrect quadrant signs

Check whether the trigonometric function is positive or negative in each quadrant.

Confusing degrees and radians

Use consistent angle units throughout the calculation.

Forgetting to adjust the interval for multiple angles

For sin 2x, the angle 2x covers twice the interval of x.

Dividing by a trigonometric expression that may equal zero

This can eliminate valid solutions.

Failing to factor quadratic expressions

Factoring often reveals multiple possible trigonometric values.

Accepting values outside the range of sine or cosine

Sine and cosine cannot exceed 1 or fall below −1.

Squaring without checking for extraneous solutions

Squaring can introduce values that do not satisfy the original equation.

Forgetting interval endpoints

An endpoint may be included or excluded depending on the inequality symbols.

Rounding too early

Keep sufficient precision until the final answer.

Not verifying solutions

Substitution into the original equation helps identify mistakes.

Key Terms

Trigonometric Equation: An equation containing one or more trigonometric functions of an unknown angle.

Solution: A value of the variable that makes an equation true.

General Solution: A formula representing all solutions to a periodic equation.

Specified Interval: The range of values within which solutions must be found.

Periodic Function: A function whose values repeat at regular intervals.

Period: The interval required for a function to complete one cycle.

Reference Angle: The acute angle between an angle's terminal side and the x-axis.

Principal Value: The unique angle returned by an inverse trigonometric function within its defined range.

Inverse Sine: The function arcsin(x), used to determine an angle from its sine value.

Inverse Cosine: The function arccos(x), used to determine an angle from its cosine value.

Inverse Tangent: The function arctan(x), used to determine an angle from its tangent value.

Quadrant: One of the four regions of the Cartesian coordinate plane.

Unit Circle: A circle of radius 1 centred at the origin.

Coterminal Angles: Angles that differ by complete revolutions and share the same terminal side.

Trigonometric Identity: An equation involving trigonometric functions that is true wherever both sides are defined.

Pythagorean Identity: A trigonometric relationship derived from sin² x + cos² x = 1.

Double-Angle Identity: An identity expressing a trigonometric function of 2x using functions of x.

Factoring: Rewriting an expression as a product of factors.

Quadratic Trigonometric Equation: A trigonometric equation containing a squared trigonometric function that can be treated like a quadratic equation.

Substitution: Replacing an expression with another variable or equivalent expression to simplify an equation.

Extraneous Solution: A value obtained during solving that does not satisfy the original equation.

Verification: Checking a proposed solution by substituting it into the original equation.

Graphical Solution: A solution found by identifying intersections of graphs.

Phase Shift: A horizontal translation of a periodic function.

Amplitude: The maximum displacement of a sinusoidal function from its midline.

Angular Frequency: The rate of angular change, commonly represented by ω.

Simple Harmonic Motion: Oscillatory motion that can be modelled using sine or cosine functions.

Vector Component: The part of a vector acting along a specified coordinate direction.

Check Your Understanding

1. Explain why trigonometric equations may have multiple solutions.

2. Solve sin x = 1/2 for 0° ≤ x < 360°.

3. Solve cos x = −√3/2 for 0° ≤ x < 360°.

4. Solve tan x = −1 for 0° ≤ x < 360°.

5. Solve sin x = √2/2 for 0 ≤ x < 2π.

6. Write the general solution of tan x = √3 in degrees.

7. Solve 3sin x − 2 = 0 for 0° ≤ x < 360°. Give answers to two decimal places.

8. Explain why cos x = 1.4 has no real solutions.

9. Solve sin 2x = √3/2 for 0° ≤ x < 360°.

10. Solve cos 3x = 1 for 0° ≤ x < 360°.

11. Solve sin(x − 45°) = √2/2 for 0° ≤ x < 360°.

12. Solve sin² x − sin x = 0 for 0° ≤ x < 360°.

13. Solve 2cos² x − 3cos x + 1 = 0 for 0° ≤ x < 360°.

14. Solve sin 2x = cos x for 0° ≤ x < 360°. Verify every solution.

15. A vibrating object has displacement x(t) = 8sin(πt/3), where x is measured in centimetres and t in seconds. Determine all times between 0 and 6 seconds when the displacement is 4 cm.

Key Takeaways

• Trigonometric equations involve unknown angles within trigonometric functions.

• Trigonometric equations often have multiple solutions because sine, cosine, and tangent are periodic.

• Sine and cosine repeat every 360°, or 2π radians.

• Tangent repeats every 180°, or π radians.

• Inverse trigonometric functions provide principal angles, but additional solutions may exist.

• Reference angles help identify solutions in different quadrants.

• The unit circle provides a visual method for locating trigonometric solutions.

• General solutions use an integer parameter to represent infinitely many angles.

• A specified interval determines which solutions must be included.

• Equations involving coefficients should first be rearranged to isolate the trigonometric function.

• Sine and cosine equations have no real solutions when the required function value lies outside [−1, 1].

• Multiple-angle equations require careful adjustment of the angle interval.

• Horizontal translations must be accounted for when solving equations involving phase shifts.

• Pythagorean and double-angle identities can simplify trigonometric equations.

• Factoring is useful for equations involving products and powers of trigonometric functions.

• Quadratic trigonometric equations can often be solved using substitution.

• Dividing by a trigonometric factor may eliminate valid solutions.

• Squaring equations can introduce extraneous solutions.

• Graphical methods can help estimate solutions and verify algebraic results.

• Every proposed solution should be checked against the original equation.

• Trigonometric equations are used in wave motion, oscillations, rotating machinery, navigation, and vector analysis.

• Mastering trigonometric equations provides an essential foundation for advanced trigonometry, mathematical modelling, physics, engineering, and calculus.

5. Proofs with Trigonometric Identities

Proofs with Trigonometric Identities

A trigonometric identity is an equation that is true for every angle for which both sides are defined. Proving a trigonometric identity involves using established mathematical relationships and algebraic reasoning to demonstrate that two expressions are equivalent.

For example:

sin² θ + cos² θ = 1

is a fundamental trigonometric identity.

However, more complicated identities may involve fractions, products, multiple angles, or several different trigonometric functions.

Consider:

(1 − cos² θ)/sin θ = sin θ

At first glance, the two sides appear different. However, applying the Pythagorean identity allows us to show that they are equivalent wherever the original expression is defined.

Proving trigonometric identities develops mathematical reasoning, algebraic fluency, and the ability to communicate logical arguments clearly. These skills are essential in advanced mathematics, calculus, physics, and engineering.

What Does It Mean to Prove an Identity?

A mathematical proof is a logical sequence of justified steps demonstrating that a statement is true.

In trigonometry, an identity proof shows that two expressions have the same value for every angle in their common domain.

For example:

tan θ cos θ = sin θ

We can verify this relationship by using the quotient identity:

tan θ = sin θ/cos θ

Starting with the left-hand side:

tan θ cos θ

= (sin θ/cos θ)cos θ

= sin θ

Therefore:

tan θ cos θ = sin θ

The identity is verified wherever cos θ ≠ 0.

Identities Compared with Equations

An identity is true for all permitted values of the variable.

An equation may only be true for certain values.

For example:

sin θ = 1/2

is an equation because it is only true for particular angles.

However:

sin² θ + cos² θ = 1

is an identity because it is true for every real angle.

The goal of an identity proof is not to find an angle. It is to establish an equivalent mathematical relationship.

The Fundamental Identities Used in Proofs

Most trigonometric proofs rely on a relatively small collection of fundamental identities.

Reciprocal Identities

csc θ = 1/sin θ

sec θ = 1/cos θ

cot θ = 1/tan θ

These identities can also be rearranged:

sin θ = 1/csc θ

cos θ = 1/sec θ

tan θ = 1/cot θ

Quotient Identities

tan θ = sin θ/cos θ

cot θ = cos θ/sin θ

These are especially useful because they allow expressions involving tangent and cotangent to be rewritten using sine and cosine.

Pythagorean Identities

sin² θ + cos² θ = 1

1 + tan² θ = sec² θ

1 + cot² θ = csc² θ

Useful rearrangements include:

sin² θ = 1 − cos² θ

cos² θ = 1 − sin² θ

tan² θ = sec² θ − 1

cot² θ = csc² θ − 1

Double-Angle Identities

sin 2θ = 2sin θ cos θ

cos 2θ = cos² θ − sin² θ

cos 2θ = 2cos² θ − 1

cos 2θ = 1 − 2sin² θ

These identities are useful when expressions contain squared trigonometric functions or multiple angles.

Understanding the Structure of an Identity Proof

A complete identity proof usually contains four parts.

Statement: Identify the identity that must be proved.

Starting expression: Begin with one side of the proposed identity.

Justified transformations: Apply known identities and valid algebraic operations.

Conclusion: Show that the starting expression becomes the other side.

Example

Verify:

sec θ cos θ = 1

Start with the left-hand side:

sec θ cos θ

= (1/cos θ)cos θ — Reciprocal identity

= 1 — Cancel the common factor cos θ

Therefore:

sec θ cos θ = 1

The identity is verified for cos θ ≠ 0.

Notice that every step has a mathematical justification.

This is more convincing than simply writing the final result.

Why We Usually Work with One Side

A common strategy is to begin with the more complicated side and simplify it until it matches the other side.

Consider:

(1 − sin² θ)/cos θ = cos θ

The left-hand side is more complicated because it contains a fraction and a squared trigonometric function.

Start with:

(1 − sin² θ)/cos θ

Use:

1 − sin² θ = cos² θ

Then:

= cos² θ/cos θ

Simplify:

= cos θ

The identity is verified wherever cos θ ≠ 0.

Why Not Change Both Sides?

Manipulating both sides independently can make a proof difficult to follow.

It may also lead to circular reasoning, where the identity being proved is assumed to be true.

A clearer approach is:

Start with one expression → Apply established identities → Reach the other expression

This creates a logical chain of equivalent expressions.

Justifying Every Algebraic Step

Each step in a mathematical proof should follow from a valid rule.

Common justifications include:

Example of a Clearly Justified Proof

Verify:

(sec² θ − 1)cos² θ = sin² θ

Therefore, the identity is verified wherever cos θ ≠ 0.

Recognising Useful Identity Substitutions

One of the most important proof-writing skills is recognising which identity to use.

Different expressions suggest different substitutions.

Expressions Containing sec² θ − 1

Recall:

1 + tan² θ = sec² θ

Therefore:

sec² θ − 1 = tan² θ

Expressions Containing 1 − cos² θ

Recall:

sin² θ + cos² θ = 1

Therefore:

1 − cos² θ = sin² θ

Expressions Containing 1 + cot² θ

Use:

1 + cot² θ = csc² θ

Expressions Containing 2sin θ cos θ

Use:

2sin θ cos θ = sin 2θ

Expressions Containing cos² θ − sin² θ

Use:

cos² θ − sin² θ = cos 2θ

Recognising these patterns reduces the number of algebraic steps required.

Proofs Using Reciprocal Identities

Reciprocal identities are often the simplest starting point.

Worked Example

Verify:

sin θ csc θ = 1

Start with:

sin θ csc θ

Replace cosecant:

= sin θ(1/sin θ)

Cancel the common factor:

= 1

The identity is verified for sin θ ≠ 0.

Worked Example

Verify:

tan θ cot θ = 1

Start with:

tan θ cot θ

Replace cotangent:

= tan θ(1/tan θ)

Simplify:

= 1

This is valid wherever both tangent and cotangent are defined, requiring sin θ ≠ 0 and cos θ ≠ 0.

Important Observation

Reciprocal identities frequently allow expressions to simplify to 1.

However, the relevant trigonometric functions must be defined.

Proofs Using Quotient Identities

Quotient identities are useful when an expression contains tangent or cotangent.

Worked Example

Verify:

tan θ cos θ = sin θ

Start with:

tan θ cos θ

Replace tangent:

= (sin θ/cos θ)cos θ

Cancel cos θ:

= sin θ

The identity is verified wherever cos θ ≠ 0.

Worked Example

Verify:

cot θ sin θ = cos θ

Start with:

cot θ sin θ

Replace cotangent:

= (cos θ/sin θ)sin θ

Cancel sin θ:

= cos θ

The identity is verified wherever sin θ ≠ 0.

When Should We Convert to Sine and Cosine?

Converting expressions into sine and cosine is especially useful when:

• Several different trigonometric functions appear.

• Fractions contain tangent, secant, or cosecant.

• The expression contains products that may cancel.

• It is unclear which Pythagorean identity to apply.

Because sine and cosine form the foundation of the other trigonometric functions, this approach often simplifies complicated proofs.

Proofs Using Pythagorean Identities

Pythagorean identities are particularly useful when expressions contain squared trigonometric functions.

Worked Example

Verify:

1 − cos² θ = sin² θ

Start with the fundamental identity:

sin² θ + cos² θ = 1

Subtract cos² θ:

sin² θ = 1 − cos² θ

Therefore:

1 − cos² θ = sin² θ

The identity is verified for every real θ.

Worked Example

Verify:

sec² θ − tan² θ = 1

Start with:

sec² θ − tan² θ

Use:

sec² θ = 1 + tan² θ

Substitute:

= 1 + tan² θ − tan² θ

Simplify:

= 1

The identity is verified wherever cos θ ≠ 0.

Combining Reciprocal and Pythagorean Identities

More advanced proofs often require multiple substitutions.

Worked Example

Verify:

(1 − cos² θ)csc² θ = 1

Start with the left-hand side:

(1 − cos² θ)csc² θ

Use the Pythagorean identity:

= sin² θ csc² θ

Replace cosecant:

= sin² θ(1/sin² θ)

Simplify:

= 1

Therefore, the identity is verified wherever sin θ ≠ 0.

Worked Example

Verify:

(sec² θ − 1)/sec² θ = sin² θ

Start with:

(sec² θ − 1)/sec² θ

Use:

sec² θ − 1 = tan² θ

Then:

= tan² θ/sec² θ

Replace tangent and secant:

= (sin² θ/cos² θ)/(1/cos² θ)

Multiply by the reciprocal:

= (sin² θ/cos² θ)cos² θ

Simplify:

= sin² θ

The identity is verified wherever cos θ ≠ 0.

Using Common Denominators in Proofs

When an expression contains fractions, a common denominator may reveal a useful identity.

Worked Example

Verify:

1/(1 − sin θ) + 1/(1 + sin θ) = 2sec² θ

Start with the left-hand side:

1/(1 − sin θ) + 1/(1 + sin θ)

Find a common denominator:

= [(1 + sin θ) + (1 − sin θ)]/[(1 − sin θ)(1 + sin θ)]

Simplify the numerator:

= 2/[(1 − sin θ)(1 + sin θ)]

Use the difference of squares:

= 2/(1 − sin² θ)

Apply the Pythagorean identity:

= 2/cos² θ

Use the reciprocal identity:

= 2sec² θ

Therefore, the identity is verified wherever cos θ ≠ 0.

Why This Works

The common denominator creates:

1 − sin² θ

which can be replaced with:

cos² θ

Recognising this structure is a valuable proof-writing skill.

Factoring Trigonometric Expressions

Factoring can reveal identities that are difficult to recognise in expanded form.

Recall the difference-of-squares identity:

a² − b² = (a − b)(a + b)

Worked Example

Verify:

(1 − cos θ)(1 + cos θ) = sin² θ

Start with:

(1 − cos θ)(1 + cos θ)

Apply the difference of squares:

= 1 − cos² θ

Use the Pythagorean identity:

= sin² θ

The identity is verified for every real θ.

Worked Example

Verify:

(sec θ − tan θ)(sec θ + tan θ) = 1

Start with:

(sec θ − tan θ)(sec θ + tan θ)

Apply the difference of squares:

= sec² θ − tan² θ

Use:

sec² θ − tan² θ = 1

Therefore:

= 1

The identity is verified wherever cos θ ≠ 0.

Proofs Involving Rationalisation

Sometimes an expression contains a denominator such as:

1 + cos θ

or:

1 − sin θ

Multiplying by the conjugate can create a difference of squares.

Worked Example

Verify:

sin θ/(1 + cos θ) = (1 − cos θ)/sin θ

Start with the left-hand side:

sin θ/(1 + cos θ)

Multiply numerator and denominator by the conjugate:

= [sin θ(1 − cos θ)]/[(1 + cos θ)(1 − cos θ)]

Use the difference of squares:

= [sin θ(1 − cos θ)]/(1 − cos² θ)

Apply the Pythagorean identity:

= [sin θ(1 − cos θ)]/sin² θ

Cancel one factor of sin θ:

= (1 − cos θ)/sin θ

Therefore, the identity is verified wherever sin θ ≠ 0.

Important Principle

Multiplying the numerator and denominator by the same nonzero expression does not change the value of a fraction.

However, we must ensure that the expression used is not zero.

Proofs Involving Double-Angle Identities

Double-angle identities are useful when expressions contain 2θ or squared trigonometric functions.

Recall:

sin 2θ = 2sin θ cos θ

cos 2θ = 1 − 2sin² θ

cos 2θ = 2cos² θ − 1

Worked Example

Verify:

sin 2θ/(2cos θ) = sin θ

Start with:

sin 2θ/(2cos θ)

Use the sine double-angle identity:

= 2sin θ cos θ/(2cos θ)

Cancel common factors:

= sin θ

The identity is verified wherever cos θ ≠ 0.

Worked Example

Verify:

1 − cos 2θ = 2sin² θ

Start with:

1 − cos 2θ

Use:

cos 2θ = 1 − 2sin² θ

Substitute:

= 1 − (1 − 2sin² θ)

Expand:

= 1 − 1 + 2sin² θ

Simplify:

= 2sin² θ

The identity is verified for every real θ.

Proofs Combining Double-Angle and Quotient Identities

Worked Example

Verify:

sin 2θ/(1 + cos 2θ) = tan θ

Start with the left-hand side:

sin 2θ/(1 + cos 2θ)

Use:

sin 2θ = 2sin θ cos θ

and:

cos 2θ = 2cos² θ − 1

Substitute:

= 2sin θ cos θ/[1 + 2cos² θ − 1]

Simplify:

= 2sin θ cos θ/(2cos² θ)

Cancel common factors:

= sin θ/cos θ

Use the quotient identity:

= tan θ

Therefore, the identity is verified wherever cos θ ≠ 0.

Another Example

Verify:

(1 − cos 2θ)/sin 2θ = tan θ

Start with:

(1 − cos 2θ)/sin 2θ

Use:

1 − cos 2θ = 2sin² θ

and:

sin 2θ = 2sin θ cos θ

Substitute:

= 2sin² θ/(2sin θ cos θ)

Simplify:

= sin θ/cos θ

Therefore:

= tan θ

This identity is verified wherever sin θ ≠ 0 and cos θ ≠ 0, which is the domain of the original left-hand side.

Proofs Involving Sum and Difference Identities

Compound-angle identities can also be used in proofs.

Recall:

sin(A + B) = sin A cos B + cos A sin B

sin(A − B) = sin A cos B − cos A sin B

cos(A + B) = cos A cos B − sin A sin B

cos(A − B) = cos A cos B + sin A sin B

Worked Example

Verify:

sin(A + B) + sin(A − B) = 2sin A cos B

Start with:

sin(A + B) + sin(A − B)

Expand both expressions:

= sin A cos B + cos A sin B + sin A cos B − cos A sin B

Combine like terms:

= 2sin A cos B

Therefore, the identity is verified.

Worked Example

Verify:

cos(A − B) − cos(A + B) = 2sin A sin B

Start with:

cos(A − B) − cos(A + B)

Expand:

= cos A cos B + sin A sin B − (cos A cos B − sin A sin B)

Remove brackets:

= cos A cos B + sin A sin B − cos A cos B + sin A sin B

Combine:

= 2sin A sin B

The identity is verified.

Constructing Complete Identity Proofs

A complete proof should be logically organised.

Consider:

(1 + tan² θ)cos² θ = 1

A clear proof is:

Proof

Start with the left-hand side:

(1 + tan² θ)cos² θ

= sec² θ cos² θ — Pythagorean identity

= (1/cos² θ)cos² θ — Reciprocal identity

= 1 — Algebraic simplification

Therefore:

(1 + tan² θ)cos² θ = 1

The identity is verified wherever cos θ ≠ 0.

Features of a Strong Proof

A strong proof:

• Begins with a clearly identified expression.

• Uses valid mathematical relationships.

• Includes enough intermediate steps to follow the reasoning.

• Identifies the important identities used.

• Avoids unsupported algebraic jumps.

• Respects domain restrictions.

• Ends with a clear conclusion.

Recognising Which Side to Simplify

Sometimes the left-hand side is more complicated.

For example:

(1 − sin² θ)/cos θ = cos θ

The left-hand side is the natural starting point.

In other cases, the right-hand side may be more complicated.

Consider:

sin θ = tan θ cos θ

We can begin with the right-hand side:

tan θ cos θ

= (sin θ/cos θ)cos θ

= sin θ

The identity is verified wherever cos θ ≠ 0.

General Guideline

Choose the side that contains:

• More terms.

• More fractions.

• More different trigonometric functions.

• Squared expressions that suggest Pythagorean identities.

• Products that may simplify.

• Expressions that can be factored.

The goal is to make the proof as clear and efficient as possible.

Recognising Common Patterns

Experienced problem-solvers learn to recognise patterns rather than memorising a separate method for every proof.

These patterns provide useful starting points for many identity proofs.

Using Algebraic Substitution

Some proofs become easier when a trigonometric expression is temporarily represented by another variable.

Worked Example

Verify:

sin⁴ θ − cos⁴ θ = sin² θ − cos² θ

Start with:

sin⁴ θ − cos⁴ θ

Recognise a difference of squares:

= (sin² θ − cos² θ)(sin² θ + cos² θ)

Use:

sin² θ + cos² θ = 1

Therefore:

= (sin² θ − cos² θ)(1)

= sin² θ − cos² θ

The identity is verified for every real θ.

Why This Is Useful

The expression:

sin⁴ θ − cos⁴ θ

may initially look complicated.

However, recognising:

a² − b² = (a − b)(a + b)

reveals a Pythagorean identity.

This illustrates how algebra and trigonometry work together.

Advanced Proof: Combining Several Techniques

Verify:

(1 + sin θ)/cos θ = cos θ/(1 − sin θ)

Start with the left-hand side:

(1 + sin θ)/cos θ

Multiply numerator and denominator by:

1 − sin θ

Then:

= [(1 + sin θ)(1 − sin θ)]/[cos θ(1 − sin θ)]

Use the difference of squares:

= (1 − sin² θ)/[cos θ(1 − sin θ)]

Apply the Pythagorean identity:

= cos² θ/[cos θ(1 − sin θ)]

Cancel one factor of cosine:

= cos θ/(1 − sin θ)

Therefore, the identity is verified wherever cos θ ≠ 0.

What Makes This Proof More Challenging?

It requires several techniques:

• Recognising the conjugate.

• Multiplying by an equivalent form of 1.

• Applying the difference of squares.

• Using a Pythagorean identity.

• Cancelling common factors.

A good proof combines these techniques in a logical sequence.

Avoiding Circular Reasoning

Circular reasoning occurs when we assume the identity we are trying to prove.

For example, suppose we want to verify:

sin² θ + cos² θ = 1

It would not be a valid independent proof to begin by assuming:

sin² θ + cos² θ = 1

and then conclude that the identity is true.

However, when verifying another identity, we may use this relationship because it is already established.

Correct Reasoning

Suppose we want to verify:

1 − sin² θ = cos² θ

We may begin with the established Pythagorean identity:

sin² θ + cos² θ = 1

Rearranging gives:

cos² θ = 1 − sin² θ

This is valid because the proof uses a previously established relationship.

Important Principle

A proof must rely on established facts, definitions, and valid operations rather than assuming the statement being proved.

Domain Restrictions in Identity Proofs

Trigonometric identities are only valid where their expressions are defined.

Consider:

tan θ cos θ = sin θ

Although the right-hand side is defined for every real angle, the left-hand side is undefined when:

cos θ = 0

Therefore, the identity is valid only where:

cos θ ≠ 0

Example

At θ = 90°:

tan 90° is undefined.

Therefore:

tan 90° cos 90°

is undefined.

We cannot treat the expression as equal to sin 90° simply because the simplified expression is defined.

Common Domain Restrictions

When cancelling a factor, we must also ensure that the factor is not zero.

Checking Identities Numerically

Numerical substitution can be useful for checking a proposed identity.

For example:

1 + tan² θ = sec² θ

At θ = 45°:

Left-hand side:

1 + tan² 45°

= 1 + 1²

= 2

Right-hand side:

sec² 45°

= (√2)²

= 2

Both sides agree.

However, checking one angle does not prove an identity.

Even checking several angles does not establish that a relationship is always true.

Counterexample

Suppose someone claims:

sin(A + B) = sin A + sin B

Test:

A = 30°

B = 60°

Left-hand side:

sin 90° = 1

Right-hand side:

sin 30° + sin 60°

= 1/2 + √3/2

These values are not equal.

Therefore, the proposed identity is false.

A single counterexample can disprove an identity, but numerical examples cannot establish a complete proof.

Graphical Interpretation of Trigonometric Identities

A true trigonometric identity produces the same output values on both sides wherever both expressions are defined.

For example:

2sin x cos x = sin 2x

If both expressions are graphed, their curves overlap.

Graphical Verification

Graphs can help students:

• Recognise equivalent expressions.

• Identify possible errors.

• Compare periodic behaviour.

• Investigate whether a proposed identity is plausible.

• Locate points where an expression is undefined.

However, graphs do not replace algebraic proofs.

A graph may suggest that two expressions are equivalent, but a mathematical proof establishes the relationship.

Applications in Physics: Resolving Forces

Trigonometric identities are useful when working with vectors.

Suppose a force F acts at an angle θ above the horizontal.

Its components are:

Fₓ = F cos θ

Fᵧ = F sin θ

The magnitude of the force can be calculated using:

F² = Fₓ² + Fᵧ²

Substitute:

Fₓ² + Fᵧ²

= F²cos² θ + F²sin² θ

Factor:

= F²(cos² θ + sin² θ)

Apply the Pythagorean identity:

= F²(1)

= F²

Therefore:

Fₓ² + Fᵧ² = F²

This proof confirms that the vector components are consistent with the magnitude of the original force.

Applications in Circular Motion

The position of an object moving in a circle of radius r can be written:

x = r cos θ

y = r sin θ

We can prove that the object remains on the circle.

Start with:

x² + y²

Substitute:

= r²cos² θ + r²sin² θ

Factor:

= r²(cos² θ + sin² θ)

Apply the Pythagorean identity:

= r²(1)

Therefore:

x² + y² = r²

This proves that the object's distance from the origin remains equal to r.

Applications in Waves and Oscillations

Trigonometric identities are useful for analysing periodic signals.

Consider:

sin(ωt + φ)

Using the sine addition identity:

sin(ωt + φ)

= sin(ωt)cos φ + cos(ωt)sin φ

This proves that a phase-shifted sine wave can be represented as a combination of sine and cosine waves.

Such relationships are useful in:

• Sound wave analysis.

• Alternating-current circuits.

• Mechanical vibrations.

• Signal processing.

• Electromagnetic waves.

A Strategy for Constructing Trigonometric Proofs

When attempting a trigonometric identity proof, use the following process.

1. Read the identity carefully. Identify the expressions on both sides.

2. Check the domain. Determine where the expressions are defined.

3. Choose a starting side. Usually select the more complicated expression.

4. Look for familiar patterns. Identify reciprocal, quotient, Pythagorean, double-angle, or compound-angle identities.

5. Rewrite expressions strategically. Converting to sine and cosine is often useful.

6. Apply algebraic techniques. Factor, expand, combine fractions, or multiply by conjugates.

7. Justify each step. State the identity or algebraic rule used.

8. Avoid unnecessary manipulation. Keep the proof focused.

9. Reach the other side. Show that the expressions are equivalent.

10. Write a conclusion. State that the identity has been verified on its domain.

When a Proof Becomes Difficult

If the expression is not simplifying, consider:

• Replacing tangent, secant, cotangent, and cosecant with sine and cosine.

• Looking for a Pythagorean identity.

• Finding a common denominator.

• Factoring a numerator or denominator.

• Multiplying by a conjugate.

• Using a double-angle identity in reverse.

• Beginning with the other side instead.

The most efficient proof is not necessarily the one with the fewest written steps. It is the one with the clearest valid reasoning.

Common Mistakes

Assuming the identity is true without proving it

A proof must demonstrate why the relationship holds.

Changing both sides without clear justification

It is generally clearer to simplify one side into the other.

Using an incorrect identity

For example, sin² θ + cos² θ = 1, not sin θ + cos θ = 1.

Confusing sin² θ with sin 2θ

These expressions have different meanings.

Cancelling terms that are not common factors

Cancellation is valid for factors, not arbitrary terms in sums or differences.

Dividing by zero

Always check whether a cancelled or divided expression may equal zero.

Ignoring domain restrictions

Equivalent simplified expressions may be defined at points where the original expression is not.

Expanding expressions incorrectly

Be careful when expanding squared binomials and products.

Forgetting brackets

For example:

1 − (1 − 2sin² θ)

must become:

2sin² θ

Using numerical examples as proof

Numerical agreement does not establish an identity.

Skipping important steps

A complete proof should show how each expression follows from the previous one.

Using circular reasoning

Do not assume the statement being proved.

Failing to recognise useful substitutions

Expressions such as 1 − sin² θ and sec² θ − 1 suggest familiar identities.

Not stating a conclusion

End by identifying the established equality and its domain.

Key Terms

Trigonometric Identity: An equation involving trigonometric functions that is true wherever both sides are defined.

Identity Proof: A logical demonstration that two trigonometric expressions are equivalent.

Verification: The process of confirming a mathematical identity using established relationships.

Mathematical Reasoning: The use of logical arguments and valid mathematical rules to establish conclusions.

Algebraic Justification: An explanation of the rule that permits a mathematical transformation.

Equivalent Expressions: Expressions that produce the same values over their common domain.

Reciprocal Identity: A relationship connecting a trigonometric function with its reciprocal.

Quotient Identity: An identity expressing tangent or cotangent using sine and cosine.

Pythagorean Identity: A trigonometric identity derived from the relationship sin² θ + cos² θ = 1.

Double-Angle Identity: An identity expressing a trigonometric function of 2θ using functions of θ.

Sum and Difference Identity: An identity relating a trigonometric function of combined angles to the functions of individual angles.

Substitution: Replacing an expression with an equivalent expression.

Simplification: Rewriting an expression in a less complicated or more useful form.

Factorisation: Expressing an algebraic expression as a product of factors.

Expansion: Removing brackets by applying multiplication and the distributive property.

Common Denominator: A shared denominator used when combining fractions.

Difference of Squares: The algebraic relationship a² − b² = (a − b)(a + b).

Conjugate: An expression formed by changing the sign between two terms of a binomial.

Rationalisation: Rewriting a fraction to remove an unwanted radical or simplify a denominator.

Cancellation: Removing a common nonzero factor from the numerator and denominator of a fraction.

Domain Restriction: A condition excluding values for which an expression is undefined.

Circular Reasoning: An invalid argument that assumes the conclusion it is supposed to prove.

Counterexample: A specific example demonstrating that a proposed general statement is false.

Unit Circle: A circle of radius 1 centred at the origin.

Vector Component: The horizontal or vertical part of a vector.

Phase Shift: A horizontal displacement of a periodic function.

Sinusoidal Function: A function represented using sine or cosine.

Periodic Function: A function whose values repeat at regular intervals.

Logical Conclusion: A statement that follows from previously established facts and valid reasoning.

Check Your Understanding

1. Explain the difference between verifying a trigonometric identity and solving a trigonometric equation.

2. Explain why it is usually helpful to begin with the more complicated side of an identity.

3. Verify:

sin θ csc θ = 1

State the domain restriction.

4. Verify:

tan θ cos θ = sin θ

5. Verify:

1 − sin² θ = cos² θ

6. Verify:

sec² θ − tan² θ = 1

7. Verify:

(1 − cos² θ)/sin θ = sin θ

8. Verify:

(1 − cos θ)(1 + cos θ) = sin² θ

9. Verify:

(sec θ − tan θ)(sec θ + tan θ) = 1

10. Verify:

(1 + tan² θ)cos² θ = 1

11. Verify:

sin 2θ/(2cos θ) = sin θ

12. Verify:

(1 − cos 2θ)/sin 2θ = tan θ

State the domain restrictions.

13. Verify:

sin(A + B) + sin(A − B) = 2sin A cos B

14. Verify:

1/(1 − sin θ) + 1/(1 + sin θ) = 2sec² θ

Show each step and justify the substitutions used.

15. Construct a complete proof of:

(1 + sin θ)/cos θ = cos θ/(1 − sin θ)

Include the identity substitutions, algebraic justifications, domain restrictions, and a concluding statement.

Key Takeaways

• A trigonometric identity proof demonstrates that two expressions are equivalent wherever both sides are defined.

• Proving an identity is different from solving a trigonometric equation.

• A complete proof uses established identities and valid algebraic operations.

• It is usually most effective to simplify one side until it matches the other.

• Every important algebraic step should have a clear justification.

• Reciprocal identities are useful for expressions involving secant, cosecant, and cotangent.

• Quotient identities allow tangent and cotangent to be rewritten using sine and cosine.

• Pythagorean identities are particularly useful when squared trigonometric functions appear.

• Double-angle identities help simplify expressions involving 2θ.

• Sum and difference identities can be used to prove relationships involving compound angles.

• Rewriting expressions entirely in sine and cosine is often an effective strategy.

• Common denominators can reveal useful Pythagorean relationships.

• Factoring and the difference of squares are important techniques in trigonometric proofs.

• Multiplying by a conjugate can simplify complicated fractions.

• Cancellation is only valid when the cancelled factor is nonzero.

• Domain restrictions must be considered throughout a proof.

• Circular reasoning must be avoided.

• Numerical examples and graphs can support understanding but cannot replace a complete mathematical proof.

• Clear notation, logical organisation, and justified steps make mathematical reasoning easier to communicate.

• Trigonometric proofs are useful in vector analysis, circular motion, waves, and oscillations.

• Mastering trigonometric identity proofs strengthens algebraic reasoning and prepares students for advanced mathematics, calculus, and mathematical modelling.