2. Compound Angle Identities

Compound Angle Identities

Compound angle identities are trigonometric formulas that allow us to calculate the sine, cosine, and tangent of angles formed by adding or subtracting two other angles.

For example, 75° is not one of the standard special angles commonly memorised in trigonometry. However, it can be expressed as:

75° = 45° + 30°

Because the exact trigonometric values of 45° and 30° are known, we can use compound angle identities to calculate sin 75°, cos 75°, and tan 75° exactly.

Compound angle identities are also important for simplifying complicated expressions, proving trigonometric relationships, solving equations, and analysing rotations, waves, and periodic motion.

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 identities express trigonometric functions of these combined angles using functions of the individual angles.

For example:

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

This means that we can calculate sin(A + B) without directly measuring or evaluating the combined angle.

Review of Special-Angle Values

These exact values are the foundation for many compound-angle calculations.

The Sine Addition Identity

The sine addition identity is:

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

This identity contains two products:

• The sine of the first angle multiplied by the cosine of the second.

• The cosine of the first angle multiplied by the sine of the second.

The two products are then added.

Worked Example: Finding sin 75°

Express:

75° = 45° + 30°

Apply the identity:

sin 75° = 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

Approximately:

sin 75° ≈ 0.9659

The exact answer is preferable when the problem asks for an exact value.

The Sine Difference Identity

The sine difference identity is:

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

The structure is almost identical to the addition identity, but the two products are subtracted.

Worked Example: Finding sin 15°

Express:

15° = 45° − 30°

Apply:

sin 15° = sin(45° − 30°)

= sin 45° cos 30° − cos 45° sin 30°

Substitute:

= (√2/2)(√3/2) − (√2/2)(1/2)

Simplify:

= √6/4 − √2/4

Therefore:

sin 15° = (√6 − √2)/4

Approximately:

sin 15° ≈ 0.2588

Important Observation

Compare:

sin 75° = (√6 + √2)/4

sin 15° = (√6 − √2)/4

The expressions differ only in the sign between the radicals.

This follows directly from the difference between the sine addition and subtraction identities.

The Cosine Addition Identity

The cosine addition identity is:

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

Notice that the cosine addition identity contains subtraction.

This is different from the sine addition identity.

Worked Example: Finding cos 75°

Express:

75° = 45° + 30°

Apply:

cos 75° = 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

This is the same exact value as sin 15°.

That makes sense because 75° and 15° are complementary angles.

The Cosine Difference Identity

The cosine difference identity is:

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

Notice that cosine subtraction uses addition.

Worked Example: Finding cos 15°

Express:

15° = 45° − 30°

Apply:

cos 15° = 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

Remembering the Signs

A useful rule is:

Sine keeps the sign; cosine changes the sign.

For sine:

• Addition uses +.

• Subtraction uses −.

For cosine:

• Addition uses −.

• Subtraction uses +.

Summary of Sine and Cosine Identities

Deriving the Tangent Addition Identity

Tangent can be expressed as:

tan θ = sin θ/cos θ

Therefore:

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

Substitute the sine and cosine addition identities:

= (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:

= [(sin A/cos A) + (sin B/cos B)]/[1 − (sin A/cos A)(sin B/cos B)]

Use the quotient identity:

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

This formula is valid where tan A and tan B are defined and the denominator is nonzero.

Worked Example: Finding tan 75°

Express:

75° = 45° + 30°

Apply:

tan 75° = (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)²/(9 − 3)

Expand:

= (12 + 6√3)/6

Therefore:

tan 75° = 2 + √3

Approximately:

tan 75° ≈ 3.7321

Deriving the Tangent Difference Identity

Begin with:

tan(A − B) = sin(A − B)/cos(A − B)

Substitute the difference identities:

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

Divide numerator and denominator by cos A cos B:

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

Worked Example: Finding tan 15°

Express:

15° = 45° − 30°

Apply:

tan 15° = (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)²/(9 − 3)

= (12 − 6√3)/6

Therefore:

tan 15° = 2 − √3

Approximately:

tan 15° ≈ 0.2679

Tangent Identity Summary

The numerator and denominator contain opposite signs.

Geometric Meaning of Compound Angle Identities

Compound angle identities can be understood through rotations in the coordinate plane.

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

(cos A, sin A)

If the point is rotated counterclockwise by an additional angle B, its new coordinates are:

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

A rotation by 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

Since the rotated point lies at angle A + B:

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

These relationships show that compound angle identities describe how coordinates change when rotations are combined.

Deriving the Difference Identities from Addition Identities

The difference identities can be derived using negative angles.

Recall:

sin(−B) = −sin B

cos(−B) = cos B

These are the odd and even properties of sine and cosine.

Deriving the Sine Difference Identity

Start with:

sin(A − B) = sin[A + (−B)]

Apply the sine addition identity:

= sin A cos(−B) + cos A sin(−B)

Use the negative-angle properties:

= sin A cos B − cos A sin B

Therefore:

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

Deriving the Cosine Difference Identity

Start with:

cos(A − B) = cos[A + (−B)]

Apply the cosine addition identity:

= cos A cos(−B) − sin A sin(−B)

Use the negative-angle properties:

= cos A cos B + sin A sin B

Therefore:

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

This shows that the difference identities are not independent formulas. They follow directly from the addition identities.

Deriving Exact Values for Other Special Angles

Compound angle identities can calculate exact values beyond the familiar 30°, 45°, and 60° angles.

Worked Example: Finding sin 105°

Express:

105° = 60° + 45°

Use:

sin 105° = sin 60° cos 45° + cos 60° sin 45°

Substitute:

= (√3/2)(√2/2) + (1/2)(√2/2)

Simplify:

sin 105° = (√6 + √2)/4

The result is positive because 105° lies in Quadrant II.

Worked Example: Finding cos 105°

Use:

cos 105° = cos 60° cos 45° − sin 60° sin 45°

Substitute:

= (1/2)(√2/2) − (√3/2)(√2/2)

Simplify:

cos 105° = (√2 − √6)/4

The result is negative, consistent with Quadrant II.

Worked Example: Finding tan 165°

Express:

165° = 120° + 45°

Use:

tan 120° = −√3

tan 45° = 1

Then:

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

This is negative, as expected in Quadrant II.

Compound Angle Identities in Radians

The identities work in radians as well as degrees.

For example:

5π/12 = π/4 + π/6

Since:

π/4 = 45°

π/6 = 30°

the combined angle is 75°.

Worked Example

Evaluate:

sin(5π/12)

Use:

sin(π/4 + π/6)

= sin(π/4)cos(π/6) + cos(π/4)sin(π/6)

Substitute:

= (√2/2)(√3/2) + (√2/2)(1/2)

Therefore:

sin(5π/12) = (√6 + √2)/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 Expressions Using Compound Angle Identities

Compound angle identities can be used in reverse.

Instead of expanding a trigonometric function of a combined angle, we can recognise a longer expression and rewrite it more simply.

Worked Example

Simplify:

sin A cos B + cos A sin B

Recognise the sine addition identity:

= sin(A + B)

Worked Example

Simplify:

cos A cos B + sin A sin B

Recognise the cosine difference identity:

= cos(A − B)

Worked Example

Simplify:

sin x cos 30° + cos x sin 30°

Apply the sine addition identity:

= sin(x + 30°)

Worked Example

Simplify:

cos x cos 45° − sin x sin 45°

Apply the cosine addition identity:

= cos(x + 45°)

Why This Is Useful

Expressions involving several products can often be rewritten as a single trigonometric function.

This makes further algebra, graphing, and equation solving easier.

Deriving Equivalent Identities

Compound angle identities can be combined to produce additional relationships.

Deriving a Sum-to-Product Relationship

Start with:

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

and:

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

Add the two equations:

sin(A + B) + sin(A − B)

= 2sin A cos B

Therefore:

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

Deriving a Difference Relationship

Subtract the sine difference identity from the sine addition identity:

sin(A + B) − sin(A − B)

= 2cos A sin B

Therefore:

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

These relationships are useful when simplifying sums of trigonometric functions.

Verifying Compound Angle Identities

Identity proofs require a sequence of valid algebraic transformations.

Worked Example

Verify:

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

Start with the left-hand side:

sin(A + B) + sin(A − B)

Expand both terms:

= 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

Simplify:

= 2sin A sin B

The identity is verified.

Deriving Double-Angle Identities from Compound Angles

Double-angle identities are special cases of compound angle identities.

Let:

A = θ

B = θ

Then:

A + B = 2θ

Sine Double-Angle Identity

Start with:

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

Substitute A = B = θ:

sin 2θ = sin θ cos θ + cos θ sin θ

Therefore:

sin 2θ = 2sin θ cos θ

Cosine Double-Angle Identity

Start with:

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

Substitute:

cos 2θ = cos² θ − sin² θ

Therefore:

cos 2θ = cos² θ − sin² θ

Using the Pythagorean identity, we also obtain:

cos 2θ = 2cos² θ − 1

cos 2θ = 1 − 2sin² θ

Tangent Double-Angle Identity

Start with:

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

Substitute A = B = θ:

tan 2θ = (tan θ + tan θ)/(1 − tan² θ)

Therefore:

tan 2θ = 2tan θ/(1 − tan² θ)

The quotient form requires tan θ to be defined and its denominator to be nonzero.

Deriving Complementary Angle Identities

Two angles are complementary if they add to 90°.

Compound angle identities can be used to prove the cofunction relationships.

Worked Example

Verify:

sin(90° − θ) = cos θ

Use the sine difference identity:

sin(90° − θ)

= sin 90° cos θ − cos 90° sin θ

Since:

sin 90° = 1

cos 90° = 0

we obtain:

= cos θ

Therefore:

sin(90° − θ) = cos θ

Worked Example

Verify:

cos(90° − θ) = sin θ

Use:

cos(90° − θ)

= cos 90° cos θ + sin 90° sin θ

= 0 + sin θ

Therefore:

cos(90° − θ) = sin θ

Solving Equations Involving Compound Angles

Compound angle identities can help solve equations containing expressions such as:

sin(x + 30°)

cos(x − 45°)

tan(2x + 15°)

These equations often involve horizontal shifts or multiple angles.

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 specified interval:

x = 0°, 120°

Verification

For x = 0°:

sin(0° + 30°) = sin 30° = 1/2

For x = 120°:

sin(120° + 30°) = sin 150° = 1/2

Both solutions satisfy the equation.

Solving Cosine Equations with Compound Angles

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°, 90°

Solving Equations Using Expanded Compound Angles

Sometimes a trigonometric equation is written as a sum of products.

Recognising a compound angle identity can simplify it.

Worked Example

Solve:

sin x cos 30° + cos x sin 30° = 1

for:

0° ≤ x < 360°

Recognise:

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

Therefore:

sin(x + 30°) = 1

Sine equals 1 at:

x + 30° = 90° + 360°n

Solve:

x = 60° + 360°n

Within the interval:

x = 60°

Worked Example

Solve:

cos x cos 60° + sin x sin 60° = 1/2

for:

0° ≤ x < 360°

Recognise the cosine difference identity:

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

Therefore:

cos(x − 60°) = 1/2

The general solutions are:

x − 60° = 60° + 360°n

or:

x − 60° = −60° + 360°n

Therefore:

x = 120° + 360°n

or:

x = 360°n

Within the interval:

x = 0°, 120°

Solving Multi-Step Compound Angle Equations

More advanced equations may combine multiple angles with algebraic operations.

Worked Example

Solve:

2sin(x + 45°) − 1 = 0

for:

0° ≤ x < 360°

First isolate sine:

2sin(x + 45°) = 1

sin(x + 45°) = 1/2

Let:

u = x + 45°

The general solutions are:

u = 30° + 360°n

or:

u = 150° + 360°n

Substitute back.

First family:

x + 45° = 30° + 360°n

x = −15° + 360°n

Within the interval:

x = 345°

Second family:

x + 45° = 150° + 360°n

x = 105° + 360°n

Within the interval:

x = 105°

Final Answer

x = 105°, 345°

Verification

For x = 105°:

2sin 150° − 1 = 2(1/2) − 1 = 0

For x = 345°:

2sin 390° − 1 = 2(1/2) − 1 = 0

Both solutions are valid.

Solving Equations Involving Multiple Angles

Worked Example

Solve:

sin(2x + 30°) = 1/2

for:

0° ≤ x < 360°

Let:

u = 2x + 30°

Since:

0° ≤ x < 360°

the transformed angle satisfies:

30° ≤ u < 750°

Sine equals 1/2 at:

u = 30°, 150°, 390°, 510°

within this interval.

Therefore:

2x + 30° = 30°, 150°, 390°, 510°

Solve each equation.

First:

2x = 0°

x = 0°

Second:

2x = 120°

x = 60°

Third:

2x = 360°

x = 180°

Fourth:

2x = 480°

x = 240°

Final Answer

x = 0°, 60°, 180°, 240°

The equation has four solutions because the argument 2x completes two full revolutions as x ranges from 0° to 360°.

Verifying Identities with Compound Angles

Compound angle identities can also be combined with Pythagorean identities.

Worked Example

Verify:

sin(x + 90°) = cos x

Start with:

sin(x + 90°)

Expand:

= sin x cos 90° + cos x sin 90°

Substitute:

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

Therefore:

= cos x

The identity is verified.

Worked Example

Verify:

cos(x + 180°) = −cos x

Start with:

cos(x + 180°)

Expand:

= cos x cos 180° − sin x sin 180°

Substitute:

= cos x(−1) − sin x(0)

Therefore:

= −cos x

The identity is verified.

Worked Example

Verify:

sin(x − 90°) = −cos x

Start with:

sin(x − 90°)

Expand:

= sin x cos 90° − cos x sin 90°

Substitute:

= sin x(0) − cos x(1)

Therefore:

= −cos x

The identity is verified.

Constructing Complete Compound Angle Proofs

A complete proof should clearly show each identity substitution and algebraic transformation.

Consider:

sin(A + B)sin(A − B) = sin² A − sin² B

Step 1: Start with the Left-Hand Side

sin(A + B)sin(A − B)

Step 2: Expand Both Sine Expressions

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

Step 3: Apply the Difference of Squares

= sin² A cos² B − cos² A sin² B

Step 4: Use Pythagorean Identities

Replace:

cos² B = 1 − sin² B

cos² A = 1 − sin² A

Then:

= sin² A(1 − sin² B) − (1 − sin² A)sin² B

Step 5: Expand

= sin² A − sin² A sin² B − sin² B + sin² A sin² B

Step 6: Combine Like Terms

The mixed terms cancel.

Therefore:

= sin² A − sin² B

The identity is verified.

This proof demonstrates how compound angle identities, algebraic expansion, and Pythagorean identities work together.

Applications in Geometry and Vector Analysis

Compound angle identities are useful when determining the components of vectors.

Suppose a vector has magnitude F and direction θ.

Its components are:

Fₓ = F cos θ

Fᵧ = F sin θ

If the vector is rotated by an additional angle φ, its new direction becomes:

θ + φ

The new components are:

Fₓ′ = F cos(θ + φ)

Fᵧ′ = F sin(θ + φ)

Using compound angle identities:

Fₓ′ = F cos θ cos φ − F sin θ sin φ

Fᵧ′ = F sin θ cos φ + F cos θ sin φ

These formulas are used in robotics, navigation, computer graphics, and engineering.

Applications in Physics: Phase-Shifted Waves

Compound angle identities are important in wave motion.

A sinusoidal wave can be written:

y(t) = A sin(ωt + φ)

where:

• A is the amplitude.

• ω is angular frequency.

• t is time.

• φ is the phase angle.

Using the sine addition identity:

sin(ωt + φ)

= sin(ωt)cos φ + cos(ωt)sin φ

Therefore:

y(t) = A sin(ωt)cos φ + A cos(ωt)sin φ

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

Worked Example

Suppose:

y(t) = 4sin(ωt + 30°)

Expand:

y(t) = 4[sin(ωt)cos 30° + cos(ωt)sin 30°]

Substitute:

= 4[(√3/2)sin(ωt) + (1/2)cos(ωt)]

Simplify:

y(t) = 2√3 sin(ωt) + 2cos(ωt)

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

Applications in Navigation

Navigation problems often involve combining directions or rotations.

Worked Example

A displacement vector has magnitude 20 km and direction 75° north of east.

Determine its exact eastward and northward components.

Eastward Component

x = 20cos 75°

Use:

cos 75° = (√6 − √2)/4

Therefore:

x = 20(√6 − √2)/4

x = 5(√6 − √2) km

Approximately:

x ≈ 5.18 km

Northward Component

y = 20sin 75°

Use:

sin 75° = (√6 + √2)/4

Therefore:

y = 5(√6 + √2) km

Approximately:

y ≈ 19.32 km

The displacement is mainly northward, which is consistent with the direction of 75° north of east.

Choosing the Correct Compound Angle Identity

Recognising these patterns helps determine which identity to apply.

A Strategy for Solving Compound Angle Problems

When working with compound angles:

1. Identify whether the problem involves addition or subtraction.

2. Determine whether sine, cosine, or tangent is required.

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

4. Select the appropriate identity.

5. Substitute exact trigonometric values.

6. Simplify fractions and radicals carefully.

7. Rationalise denominators when appropriate.

8. Check the sign using the quadrant.

9. For equations, determine all solutions in the specified interval.

10. For proofs, justify each transformation and avoid circular reasoning.

11. Verify answers using the original expression or equation.

12. State the final answer clearly.

Common Mistakes

Assuming sine distributes over addition

In general:

sin(A + B) ≠ sin A + sin B

Assuming cosine distributes over addition

In general:

cos(A + B) ≠ cos A + cos B

Confusing the signs in cosine identities

Cosine addition uses subtraction, while cosine difference uses addition.

Using the wrong tangent denominator

Tangent addition uses 1 − tan A tan B.

Tangent difference uses 1 + tan A tan B.

Forgetting the quadrant

Exact values may be negative depending on the angle.

Using incorrect special-angle values

Review the exact values for 30°, 45°, and 60°.

Mixing degrees and radians

All angles in a calculation must use consistent units.

Rounding too early

Keep exact fractions and radicals until the final answer.

Failing to rationalise denominators when required

Multiply by the appropriate conjugate.

Forgetting periodic solutions

Trigonometric equations may have multiple solutions within the specified interval.

Failing to adjust intervals for multiple angles

For expressions involving 2x or 3x, the transformed angle covers a larger interval.

Using numerical examples as complete proofs

Numerical checks are useful, but they do not establish a general identity.

Ignoring domain restrictions

The tangent addition and difference formulas require their denominators and the individual tangent values to be defined.

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 Difference 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 Difference 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 Difference 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 a commonly known exact trigonometric value.

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

Reference Angle: The acute angle 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.

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

Verification: Checking that an equation or identity is mathematically valid.

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

Double-Angle Identity: An identity expressing a trigonometric function of twice an angle.

Complementary Angles: Two angles whose sum is 90°.

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

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 simplify or remove a radical from its denominator.

Periodicity: The property of a function whose values repeat at regular intervals.

Phase Shift: A horizontal displacement of a periodic function.

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

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

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

Sinusoidal Function: A function represented using sine or cosine.

Mathematical Reasoning: The use of logical arguments and established relationships to reach valid conclusions.

Check Your Understanding

1. State the sine addition and sine difference identities.

2. State the cosine addition and cosine difference identities.

3. Derive the tangent addition identity using the sine and cosine addition identities.

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

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

6. Determine the exact value of cos 15°.

7. Determine the exact value of tan 75°.

8. Determine the exact value of sin 105°.

9. Determine the exact value of cos 165°.

10. Simplify:

sin x cos 45° + cos x sin 45°

11. Simplify:

cos x cos 30° + sin x sin 30°

12. Solve:

sin(x + 45°) = √2/2

for 0° ≤ x < 360°.

13. Solve:

2cos(x − 30°) − 1 = 0

for 0° ≤ x < 360°.

14. Verify:

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

Show each algebraic step.

15. Construct a complete proof of:

sin(A + B)sin(A − B) = sin² A − sin² B

Your proof should include the compound angle expansions, algebraic justifications, Pythagorean substitutions, and a concluding statement.

Key Takeaways

• Compound angle identities allow trigonometric functions of combined angles to be expressed using functions of the 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 difference 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 difference identity is cos(A − B) = cos A cos B + sin A sin B.

• The tangent addition and difference identities can be derived from the sine and cosine identities.

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

• Cosine identities use the opposite sign.

• Tangent identities use opposite signs in the numerator and denominator.

• Exact values for angles such as 15°, 75°, 105°, and 165° can be derived from familiar special angles.

• Compound angle identities work in both degrees and radians.

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

• The geometric meaning of compound angle identities can be understood through rotations on the unit circle.

• Difference identities can be derived from addition identities using negative-angle properties.

• Double-angle identities are special cases of compound angle identities.

• Complementary-angle relationships can be derived using sum and difference identities.

• Compound angle identities are useful for solving equations involving phase shifts and multiple angles.

• Identity proofs require valid algebraic transformations and clear mathematical reasoning.

• Quadrant signs provide an important check on calculated values.

• Compound angle identities are used in vector analysis, navigation, rotations, waves, and oscillations.

• Mastering compound angle identities provides a foundation for advanced trigonometric equations, identity proofs, calculus, and mathematical modelling.