Solving General Trigonometric Equations

Trigonometric equations are equations involving sine, cosine, tangent, or other trigonometric functions. Unlike many algebraic equations, trigonometric equations often have infinitely many solutions because trigonometric functions are periodic.

For example:

sin x = 1/2

has solutions at 30° and 150°, but these are not the only solutions. Because sine repeats every 360°, additional solutions occur at 390°, 510°, 750°, and so on.

To solve trigonometric equations successfully, we must understand how to find solutions within a specified interval, express all solutions using general formulas, and verify that the answers satisfy the original equation.

More advanced problems may involve multiple angles, compound angles, quadratic expressions, identities, or combinations of different trigonometric functions.

Understanding Trigonometric Equations

A trigonometric equation contains one or more trigonometric functions of an unknown angle.

Examples include:

• sin x = 1/2

• cos x = −√3/2

• tan x = 1

• 2sin x − 1 = 0

• sin 2x = cos x

• 2cos² x − 3cos x + 1 = 0

The goal is to determine every angle that makes the equation true within the required domain or interval.

Equations Versus Identities

A trigonometric equation is true only for particular values of the variable.

For example:

sin x = 1/2

is true at certain angles.

A trigonometric identity is true for every angle in its domain.

For example:

sin² x + cos² x = 1

This distinction is important because identities are used to transform equations into forms that are easier to solve.

Periodicity and Multiple Solutions

Trigonometric functions repeat their values at regular intervals.

The length of one complete cycle is called the period.

For example:

sin 30° = 1/2

sin 390° = 1/2

sin 750° = 1/2

Each angle differs from the previous one by 360°.

Similarly:

tan 45° = 1

tan 225° = 1

tan 405° = 1

Tangent repeats every 180°.

General Principle

If x is a solution to a sine or cosine equation, then adding or subtracting a whole number of complete periods produces another solution.

For tangent, the repeated interval is 180° rather than 360°.

This explains why general trigonometric equations often have infinitely many solutions.

Solving Equations over Specified Intervals

A specified interval limits the possible solutions.

For example:

0° ≤ x < 360°

means that x may equal 0° but must be less than 360°.

Another common interval is:

0 ≤ x < 2π

which represents one complete revolution in radians.

The endpoints matter.

For example, in:

0° ≤ x < 360°

the angle 0° is included, but 360° is excluded.

In:

0° ≤ x ≤ 360°

both endpoints are included.

Even though 0° and 360° are coterminal, they are distinct numerical values and may both need to be listed when the interval includes them.

Reference Angles and Quadrants

A reference angle is the acute angle formed between an angle's terminal side and the x-axis.

Reference angles help determine all solutions to a trigonometric equation.

For example:

sin x = 1/2

The reference angle is:

30°

Sine is positive in Quadrants I and II.

Therefore:

x = 30°

or:

x = 180° − 30° = 150°

Within one complete revolution:

x = 30°, 150°

Signs of Trigonometric Functions

A useful memory aid is ASTC:

• Quadrant I: All positive.

• Quadrant II: Sine positive.

• Quadrant III: Tangent positive.

• Quadrant IV: Cosine positive.

These signs help identify which quadrants contain valid solutions.

Solving Basic Sine Equations

Worked Example

Solve:

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 positive in Quadrants I and II.

Quadrant I:

x = 60°

Quadrant II:

x = 180° − 60° = 120°

Final Answer

x = 60°, 120°

Verification

sin 60° = √3/2

sin 120° = √3/2

Both solutions satisfy the original equation.

Solving Negative Sine Equations

Worked Example

Solve:

sin x = −1/2

for:

0° ≤ x < 360°

The reference angle is:

30°

Sine is negative in Quadrants III and IV.

Quadrant III:

x = 180° + 30° = 210°

Quadrant IV:

x = 360° − 30° = 330°

Therefore:

x = 210°, 330°

Solving Basic Cosine Equations

Worked Example

Solve:

cos x = −√2/2

for:

0° ≤ x < 360°

The reference angle is:

45°

Cosine is negative in Quadrants II and III.

Quadrant II:

x = 180° − 45° = 135°

Quadrant III:

x = 180° + 45° = 225°

Therefore:

x = 135°, 225°

Verification

cos 135° = −√2/2

cos 225° = −√2/2

Both solutions are valid.

Solving Basic Tangent Equations

Worked Example

Solve:

tan x = √3

for:

0° ≤ x < 360°

The reference angle is:

60°

Tangent is positive in Quadrants I and III.

Quadrant I:

x = 60°

Quadrant III:

x = 180° + 60° = 240°

Therefore:

x = 60°, 240°

Important Difference

Tangent has a period of 180°.

Therefore, its solutions can be written using one repeating family rather than two.

Determining General Solutions

A general solution describes all possible solutions to a trigonometric equation.

The integer n represents the number of complete periods added or subtracted.

n ∈ ℤ

This means n may be any integer:

…, −2, −1, 0, 1, 2, …

General Solutions for Sine

For:

sin x = a

where −1 ≤ a ≤ 1, let:

α = arcsin(a)

Then the general solutions in radians are:

x = α + 2πn

or:

x = π − α + 2πn

where n ∈ ℤ.

In degrees:

x = α + 360°n

or:

x = 180° − α + 360°n

where α is measured in degrees.

General Solutions for Cosine

For:

cos x = a

where −1 ≤ a ≤ 1, let:

α = arccos(a)

Then:

x = ±α + 2πn

or, in degrees:

x = ±α + 360°n

where n ∈ ℤ.

General Solutions for Tangent

For:

tan x = a

let:

α = arctan(a)

Then:

x = α + πn

or, in degrees:

x = α + 180°n

where n ∈ ℤ.

Worked Example: General Sine Solutions

Find the general solution of:

sin x = 1/2

The reference angle is:

30°

The solutions within one revolution are:

30° and 150°.

Since sine repeats every 360°:

x = 30° + 360°n

or:

x = 150° + 360°n

where n ∈ ℤ.

Interpreting the Integer n

If n = 0:

x = 30°, 150°

If n = 1:

x = 390°, 510°

If n = −1:

x = −330°, −210°

All these angles satisfy the equation.

Worked Example: General Cosine Solutions

Find the general solution of:

cos x = −1/2

Cosine equals −1/2 at:

120° and 240°.

Therefore:

x = 120° + 360°n

or:

x = 240° + 360°n

where n ∈ ℤ.

An equivalent compact form is:

x = ±120° + 360°n

because −120° is coterminal with 240°.

Worked Example: General Tangent Solutions

Find the general solution of:

tan x = −1

The reference angle is:

45°

Tangent is negative in Quadrants II and IV.

Within one revolution:

x = 135°, 315°

Since tangent repeats every 180°:

x = 135° + 180°n

where n ∈ ℤ.

An equivalent general solution is:

x = −45° + 180°n

Both describe the same infinite set of angles.

Solving Trigonometric Equations in Radians

The methods are identical when angles are measured in radians.

The important special-angle values are:

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:

x = π/4, 3π/4

General Solution

x = π/4 + 2πn

or:

x = 3π/4 + 2πn

where n ∈ ℤ.

Solving Equations Requiring Rearrangement

Many trigonometric equations must first be rearranged into a standard form.

Worked Example

Solve:

3sin x − 1 = 0

for:

0° ≤ x < 360°

Rearrange:

3sin x = 1

sin x = 1/3

Find the reference angle:

α = arcsin(1/3)

α ≈ 19.47°

Sine is positive in Quadrants I and II.

Quadrant I:

x ≈ 19.47°

Quadrant II:

x ≈ 180° − 19.47°

x ≈ 160.53°

Final Answer

x ≈ 19.47°, 160.53°

General Solution

x ≈ 19.47° + 360°n

or:

x ≈ 160.53° + 360°n

where n ∈ ℤ.

For exact general solutions, retain arcsin(1/3) rather than rounding the reference angle.

Recognising Equations with No Real Solutions

Sine and cosine have a range of:

−1 ≤ sin x ≤ 1

−1 ≤ cos x ≤ 1

Therefore:

sin x = 2

has no real solutions.

Similarly:

cos x = −1.5

has no real solutions.

Worked Example

Solve:

4cos x + 5 = 0

Rearrange:

4cos x = −5

cos x = −5/4

Since:

−5/4 = −1.25

is outside the range of cosine:

There are no real solutions.

Recognising the range can prevent unnecessary calculations.

Solving Equations Involving Multiple Angles

Expressions such as sin 2x and cos 3x complete more cycles than sin x and cos x over the same interval.

This means they may have more solutions.

Worked Example

Solve:

sin 2x = 1/2

for:

0° ≤ x < 360°

Let:

u = 2x

Then:

0° ≤ u < 720°

Solve:

sin u = 1/2

Within this expanded interval:

u = 30°, 150°, 390°, 510°

Since:

u = 2x

divide each solution by 2:

x = 15°, 75°, 195°, 255°

Final Answer

x = 15°, 75°, 195°, 255°

General Solution

From:

2x = 30° + 360°n

or:

2x = 150° + 360°n

we obtain:

x = 15° + 180°n

or:

x = 75° + 180°n

where n ∈ ℤ.

Solving Triple-Angle Equations

Worked Example

Solve:

cos 3x = 0

for:

0° ≤ x < 360°

Let:

u = 3x

Then:

0° ≤ u < 1080°

Cosine equals zero at:

u = 90°, 270°, 450°, 630°, 810°, 990°

Divide each value by 3:

x = 30°, 90°, 150°, 210°, 270°, 330°

Final Answer

x = 30°, 90°, 150°, 210°, 270°, 330°

General Solution

Cosine equals zero when:

3x = 90° + 180°n

Therefore:

x = 30° + 60°n

where n ∈ ℤ.

This single expression describes all solutions.

Solving Compound-Angle Equations

Compound-angle equations contain expressions such as:

sin(x + 30°)

cos(x − 45°)

tan(2x + 15°)

The main strategy is to isolate the complete angle expression before solving for x.

Worked Example

Solve:

sin(x − 30°) = 1/2

for:

0° ≤ x < 360°

Let:

u = x − 30°

The general solutions are:

u = 30° + 360°n

or:

u = 150° + 360°n

Substitute back:

x − 30° = 30° + 360°n

or:

x − 30° = 150° + 360°n

Solve:

x = 60° + 360°n

or:

x = 180° + 360°n

Within the specified interval:

x = 60°, 180°

Solving Equations with Both Multiple Angles and Phase Shifts

Worked Example

Solve:

cos(2x − 30°) = 1/2

for:

0° ≤ x < 360°

Let:

u = 2x − 30°

The general solutions of cos u = 1/2 are:

u = 60° + 360°n

or:

u = −60° + 360°n

First Family

2x − 30° = 60° + 360°n

2x = 90° + 360°n

x = 45° + 180°n

Within the interval:

x = 45°, 225°

Second Family

2x − 30° = −60° + 360°n

2x = −30° + 360°n

x = −15° + 180°n

Within the interval:

x = 165°, 345°

Final Answer

x = 45°, 165°, 225°, 345°

Solving Quadratic Trigonometric Equations

Some equations contain squared trigonometric functions and can be solved using algebraic factorisation.

Worked Example

Solve:

2sin² x − 3sin x + 1 = 0

for:

0° ≤ x < 360°

Let:

u = sin x

Then:

2u² − 3u + 1 = 0

Factor:

(2u − 1)(u − 1) = 0

Therefore:

u = 1/2

or:

u = 1

Substitute back.

First Case

sin x = 1/2

x = 30°, 150°

Second Case

sin x = 1

x = 90°

Final Answer

x = 30°, 90°, 150°

General Solution

x = 30° + 360°n

or:

x = 150° + 360°n

or:

x = 90° + 360°n

where n ∈ ℤ.

Solving Quadratic Cosine Equations

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

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 Pythagorean Identities

Trigonometric identities can transform equations containing more than one trigonometric function.

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 Using Double-Angle Identities

Worked Example

Solve:

sin 2x = sin x

for:

0° ≤ x < 360°

Use:

sin 2x = 2sin x cos x

Substitute:

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°

Important Algebraic Lesson

Do not divide both sides by sin x.

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

Factoring preserves all possible solutions.

Solving Equations Using Triple-Angle Identities

Worked Example

Solve:

sin 3x = sin x

for:

0° ≤ x < 360°

Use:

sin 3x = 3sin x − 4sin³ x

Substitute:

3sin x − 4sin³ x = sin x

Rearrange:

2sin x − 4sin³ x = 0

Factor:

2sin x(1 − 2sin² x) = 0

Therefore:

sin x = 0

or:

sin² x = 1/2

First Case

sin x = 0

x = 0°, 180°

Second Case

sin² x = 1/2

Therefore:

sin x = ±√2/2

The solutions are:

45°, 135°, 225°, 315°

Final Answer

x = 0°, 45°, 135°, 180°, 225°, 315°

Solving Equations Involving Sine and Cosine

Some equations contain both sine and cosine.

Worked Example

Solve:

sin x = cos x

for:

0° ≤ x < 360°

Since cosine cannot be zero at a solution of this equation, divide both sides by cos x:

sin x/cos x = 1

Use the quotient identity:

tan x = 1

Tangent equals 1 at:

45° and 225°.

Therefore:

x = 45°, 225°

General Solution

x = 45° + 180°n

where n ∈ ℤ.

Solving Equations with Coefficients of Sine and Cosine

Worked Example

Solve:

√3sin x = cos x

for:

0° ≤ x < 360°

At a solution, cos x cannot be zero, so divide both sides by cos x:

√3(sin x/cos x) = 1

Use:

tan x = sin x/cos x

Then:

√3tan x = 1

Therefore:

tan x = 1/√3

Since:

tan 30° = 1/√3

the solutions are:

x = 30°, 210°

General Solution

x = 30° + 180°n

where n ∈ ℤ.

Solving Equations Using Sum-to-Product Identities

Sum-to-product identities are particularly useful when equations contain sums or differences of trigonometric functions.

Worked Example

Solve:

sin 5x + sin 3x = 0

for:

0° ≤ x < 360°

Use:

sin A + sin B = 2sin[(A + B)/2]cos[(A − B)/2]

Therefore:

sin 5x + sin 3x

= 2sin 4x cos x

The equation becomes:

2sin 4x cos x = 0

Apply the zero-product property:

sin 4x = 0

or:

cos x = 0

First Case

sin 4x = 0

4x = 180°n

x = 45°n

Within the interval:

x = 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°

Second Case

cos x = 0

x = 90°, 270°

These are already included.

Final Answer

x = 0°, 45°, 90°, 135°, 180°, 225°, 270°, 315°

Solving Equations over Larger Intervals

Some problems specify intervals covering several revolutions.

Worked Example

Solve:

sin x = 1/2

for:

0° ≤ x < 720°

The solutions in one revolution are:

30° and 150°.

Sine repeats every 360°.

Add 360°:

390° and 510°.

Therefore:

x = 30°, 150°, 390°, 510°

Worked Example

Solve:

cos x = 0

for:

−360° ≤ x < 360°

Cosine equals zero at:

90° + 180°n

Choose integer values of n that produce angles within the interval.

The solutions are:

x = −270°, −90°, 90°, 270°

Verifying Trigonometric Solutions

Every proposed solution should satisfy the original equation.

Verification is especially important when:

• Squaring both sides.

• Multiplying by an expression containing the variable.

• Using inverse trigonometric functions.

• Applying algebraic transformations that may introduce extraneous solutions.

• Working with complicated multiple-angle equations.

Worked Example

Verify the solutions of:

2sin x − 1 = 0

The proposed solutions are:

x = 30°, 150°

For x = 30°:

2sin 30° − 1

= 2(1/2) − 1

= 0

For x = 150°:

2sin 150° − 1

= 2(1/2) − 1

= 0

Both solutions satisfy the equation.

Extraneous Solutions

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

This can happen when both sides are squared.

Worked Example

Solve:

sin x = cos² x

for:

0° ≤ x < 360°

Use:

cos² x = 1 − sin² x

Substitute:

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 ± √(1 + 4)]/2

Therefore:

u = (−1 ± √5)/2

The two algebraic values are:

u₁ = (−1 + √5)/2 ≈ 0.6180

u₂ = (−1 − √5)/2 ≈ −1.6180

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

Therefore:

sin x = (√5 − 1)/2

The reference angle is approximately:

38.17°

Since sine is positive in Quadrants I and II:

x ≈ 38.17°, 141.83°

These solutions satisfy the original equation, allowing for rounding.

The rejected algebraic root is outside the range of sine; it is not a valid trigonometric value.

Interpreting Multiple Solutions Graphically

A trigonometric equation can be interpreted as the intersection of two 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 curves intersect.

For:

0° ≤ x < 360°

there are two intersections.

For:

0° ≤ x < 720°

there are four intersections.

Why Graphs Are Useful

Graphs help us:

• Estimate the number of solutions.

• Identify repeated solutions.

• Understand the effect of changing the interval.

• Recognise the periodic nature of trigonometric functions.

• Check whether calculated answers are reasonable.

However, graphical estimates should not replace exact algebraic solutions when exact answers are required.

Applications in Physics: Oscillatory Motion

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

Suppose a mass moves according to:

x(t) = 0.20sin(2πt)

where x is displacement in metres and t is time in seconds.

Find the times during:

0 ≤ t < 2

when:

x = 0.10 m

Step 1: Form the Equation

0.20sin(2πt) = 0.10

Divide by 0.20:

sin(2πt) = 1/2

Step 2: Solve for the Angle

The general solutions are:

2πt = π/6 + 2πn

or:

2πt = 5π/6 + 2πn

Step 3: Solve for Time

Divide by 2π:

t = 1/12 + n

or:

t = 5/12 + n

Step 4: Apply the Interval

For:

0 ≤ t < 2

the solutions are:

t = 1/12, 5/12, 13/12, 17/12 seconds

Approximately:

t ≈ 0.083, 0.417, 1.083, 1.417 seconds

The object reaches the specified displacement twice during each oscillation.

Applications in Circular Motion

Suppose a point moves around a circle of radius 5 m.

Its vertical position is:

y = 5sin θ

Find all angles between 0° and 360° where:

y = 2.5 m

Substitute:

5sin θ = 2.5

Divide by 5:

sin θ = 1/2

Therefore:

θ = 30°, 150°

The point reaches the same vertical height at two different positions on the circle.

This demonstrates why multiple trigonometric solutions can represent different physical situations.

Applications in Navigation

Trigonometric equations can be used to determine possible directions.

Suppose a displacement has magnitude 100 km and an eastward component of 60 km.

The eastward component is:

x = 100cos θ

Therefore:

100cos θ = 60

Divide:

cos θ = 0.6

Find the reference angle:

θ ≈ 53.13°

For:

0° ≤ θ < 360°

cosine is positive in Quadrants I and IV.

Therefore:

θ ≈ 53.13°, 306.87°

These represent two possible directions with the same eastward component.

Additional information about the northward or southward component would be needed to choose between them.

A Strategy for Solving General Trigonometric Equations

When solving increasingly complex trigonometric equations, follow a systematic process.

1. Identify the interval. Determine whether the problem uses degrees or radians and whether endpoints are included.

2. Simplify the equation. Rearrange terms and apply algebraic rules.

3. Recognise the trigonometric structure. Look for sine, cosine, tangent, multiple angles, quadratic forms, or identities.

4. Choose an appropriate method. Use reference angles, factorisation, substitutions, or trigonometric identities.

5. Find the reference angle. Use exact special-angle values or inverse trigonometric functions.

6. Determine the correct quadrants. Use the signs of trigonometric functions.

7. Write the general solution. Include the appropriate period and integer n.

8. Apply the interval. Select every solution that satisfies the required boundaries.

9. Remove duplicates. Different branches may produce the same solution.

10. Check domain restrictions. Exclude angles where the original equation is undefined.

11. Verify solutions. Substitute answers into the original equation.

12. State the final answer clearly. Use exact values when possible and sensible rounding when necessary.

Common Mistakes

Finding only one solution

Sine and cosine equations often have two solutions per cycle.

Forgetting periodicity

General solutions require the appropriate integer multiple of the period.

Using 360° for tangent

Tangent repeats every 180°, so the most efficient general solution uses 180°n.

Ignoring the specified interval

A valid general solution may lie outside the required interval.

Including excluded endpoints

Pay attention to strict inequalities such as x < 360°.

Mixing degrees and radians

Use consistent angle units throughout a calculation.

Choosing the wrong quadrants

Use the sign of the trigonometric function to determine valid quadrants.

Forgetting negative solutions

General solutions include negative as well as positive angles.

Failing to expand the interval for multiple angles

For sin 3x, the argument covers three times the interval of x.

Dividing by a trigonometric expression that may equal zero

This can eliminate valid solutions.

Ignoring the range of sine and cosine

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

Forgetting both roots of a quadratic

Factorisation may produce multiple trigonometric equations.

Failing to verify answers

Algebraic manipulation may introduce invalid solutions.

Rounding too early

Keep exact values or sufficient decimal precision until the final answer.

Listing duplicate solutions

Different branches may produce the same angle.

Assuming every solution has the same physical meaning

Multiple solutions may represent different positions, times, or directions.

Key Terms

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

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

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

General Solution: A formula describing all solutions of an equation, usually using an integer parameter.

Principal Value: A selected value returned by an inverse trigonometric function within its defined range.

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.

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

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

Period: The length of one complete cycle of a periodic function.

Coterminal Angles: Angles that share the same terminal side.

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

Closed Interval: An interval that includes both endpoints.

Open Interval: An interval that excludes both endpoints.

General Sine Solution: A formula representing every angle with a specified sine value.

General Cosine Solution: A formula representing every angle with a specified cosine value.

General Tangent Solution: A formula representing every angle with a specified tangent value.

Inverse Sine: The function arcsin, used to determine a principal angle from a sine value.

Inverse Cosine: The function arccos, used to determine a principal angle from a cosine value.

Inverse Tangent: The function arctan, used to determine a principal angle from a tangent value.

Quadratic Trigonometric Equation: A trigonometric equation containing a squared trigonometric function.

Factorisation: Rewriting an expression as a product of factors.

Zero-Product Property: The principle that a product equals zero when at least one factor equals zero.

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

Triple-Angle Identity: An identity expressing a trigonometric function of three times an angle.

Compound-Angle Identity: An identity involving the sum or difference of two angles.

Sum-to-Product Identity: An identity converting a sum or difference of trigonometric functions into a product.

Extraneous Solution: A proposed solution obtained during manipulation that does not satisfy the original equation.

Verification: Checking that a proposed solution satisfies the original equation.

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

Range: The set of possible output values of a function.

Graphical Solution: A solution determined from the intersection of graphs.

Phase Shift: A horizontal displacement of a periodic function.

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

Oscillation: Repeated motion around an equilibrium position.

Sinusoidal Motion: Motion described using sine or cosine functions.

Mathematical Modelling: The use of mathematical relationships to represent real-world situations.

Check Your Understanding

1. Explain why trigonometric equations often have infinitely many solutions.

2. State the periods of sine, cosine, and tangent in degrees and radians.

3. Solve:

sin x = √3/2

for 0° ≤ x < 360°.

4. Solve:

cos x = −1/2

for 0° ≤ x < 360°.

5. Determine the general solution of:

tan x = 1

in degrees.

6. Determine the general solution of:

sin x = −√2/2

in radians.

7. Solve:

3sin x − 1 = 0

for 0° ≤ x < 360°. Give answers to two decimal places.

8. Explain why:

cos x = 1.5

has no real solutions.

9. Solve:

sin 2x = √3/2

for 0° ≤ x < 360°.

10. Solve:

cos 3x = 1/2

for 0° ≤ x < 360°.

11. Solve:

sin(x + 45°) = √2/2

for 0° ≤ x < 360°.

12. Solve:

2sin² x − 3sin x + 1 = 0

for 0° ≤ x < 360°.

13. Solve:

sin 2x = sin x

for 0° ≤ x < 360°.

14. Solve:

sin 5x − sin x = 0

for 0° ≤ x < 360°. Use an appropriate trigonometric identity.

15. A mass oscillates according to:

x(t) = 0.30cos(πt)

where x is measured in metres and t in seconds.

Determine every time in the interval 0 ≤ t < 4 when the displacement is 0.15 m. Explain why multiple solutions occur and verify your answers.

Key Takeaways

• Trigonometric equations are solved by finding every angle that satisfies the equation within the required domain or interval.

• Trigonometric functions are periodic, so equations often have infinitely many solutions.

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

• Tangent repeats every 180° or π radians.

• Reference angles help determine solutions in different quadrants.

• The signs of sine, cosine, and tangent determine which quadrants contain valid solutions.

• General solutions use an integer parameter n to represent repeated cycles.

• Sine equations generally have two solution families per cycle, except at their maximum and minimum values.

• Cosine equations generally have two solution families per cycle, except at their maximum and minimum values.

• Tangent equations can be represented using one repeating solution family.

• Specified intervals determine which solutions must be included in the final answer.

• Open and closed interval endpoints must be handled carefully.

• Inverse trigonometric functions provide reference angles for values that are not standard special angles.

• Sine and cosine values must lie between −1 and 1.

• Equations involving multiple angles may have additional solutions because the functions complete more cycles.

• Compound-angle equations can be solved by isolating the complete angle expression.

• Quadratic trigonometric equations can often be solved using algebraic factorisation.

• Pythagorean identities allow equations involving sine and cosine to be rewritten in a single function.

• Double-angle and triple-angle identities are useful for solving more complex equations.

• Sum-to-product identities can transform trigonometric sums into factorable equations.

• Dividing by an expression containing the variable may eliminate valid solutions.

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

• Graphs help interpret why multiple solutions occur.

• Trigonometric equations are useful in circular motion, oscillations, navigation, and other periodic systems.

• Mastering general trigonometric equations develops the algebraic reasoning, graphical understanding, and problem-solving skills needed for advanced mathematics, physics, and calculus.