Equations Involving Multiple Angles

Trigonometric equations involving multiple angles contain expressions such as sin 2x, cos 3x, tan 2x, or sin(x/2). These equations require careful attention because multiplying or dividing an angle changes the period of the trigonometric function and may change the number of solutions within a given interval.

For example, the equation:

sin 2x = 1/2

has four solutions between 0° and 360°, whereas:

sin x = 1/2

has only two solutions over the same interval.

More advanced equations may combine double-angle, half-angle, and triple-angle identities with algebraic factorisation, compound-angle formulas, and Pythagorean identities.

The key to solving these equations is to select an appropriate method, determine all possible solutions, and verify that every answer satisfies the original equation.

Understanding Equations with Multiple Angles

A multiple-angle equation contains a trigonometric function whose argument is a multiple or fraction of the unknown angle.

Examples include:

• sin 2x = √3/2

• cos 3x = 1/2

• tan 2x = 1

• sin(x/2) = 1/2

• cos 2x = sin x

• sin 3x = sin x

• 2cos² x − cos 2x = 1

There are two main approaches to solving these equations.

Direct angle solving: Treat the complete angle expression as a new variable, solve the simpler trigonometric equation, and then isolate x.

Identity substitution: Use double-angle, triple-angle, half-angle, or other trigonometric identities to rewrite the equation in terms of simpler expressions.

The most efficient approach depends on the structure of the equation.

How Multiple Angles Affect Periodicity

The basic trigonometric functions are periodic.

When the argument is multiplied by a nonzero constant k, the period changes.

For sine and cosine:

Period = 360°/|k|

or:

Period = 2π/|k|

For tangent:

Period = 180°/|k|

or:

Period = π/|k|

Examples

Multiplying the angle increases the number of cycles over a fixed interval.

Dividing the angle decreases the number of cycles.

This directly affects how many solutions may occur.

Visualising Multiple-Angle Functions

Reviewing Double-Angle Identities

The double-angle identities are derived from the compound-angle identities by setting both angles equal.

Sine

sin 2x = 2sin x cos x

Cosine

cos 2x = cos² x − sin² x

Equivalent forms include:

cos 2x = 2cos² x − 1

cos 2x = 1 − 2sin² x

Tangent

tan 2x = 2tan x/(1 − tan² x)

The tangent quotient form requires tan x to be defined and 1 − tan² x ≠ 0.

These identities are especially useful when an equation contains both x and 2x.

Solving Double-Angle Equations Directly

Worked Example: Solving sin 2x = 1/2

Solve:

sin 2x = 1/2

for:

0° ≤ x < 360°

Step 1: Introduce a New Angle

Let:

u = 2x

Since x lies between 0° and 360°:

0° ≤ u < 720°

Step 2: Solve the Sine Equation

sin u = 1/2

The solutions in the expanded interval are:

u = 30°, 150°, 390°, 510°

Step 3: Solve for x

Since u = 2x, divide each solution by 2:

x = 15°, 75°, 195°, 255°

Final Answer

x = 15°, 75°, 195°, 255°

General Solution

The general sine solutions are:

2x = 30° + 360°n

or:

2x = 150° + 360°n

Therefore:

x = 15° + 180°n

or:

x = 75° + 180°n

where n ∈ ℤ.

Solving Double-Angle Cosine Equations

Worked Example

Solve:

cos 2x = −1/2

for:

0° ≤ x < 360°

Let:

u = 2x

Then:

0° ≤ u < 720°

Cosine equals −1/2 at:

u = 120°, 240°, 480°, 600°

Divide by 2:

x = 60°, 120°, 240°, 300°

Final Answer

x = 60°, 120°, 240°, 300°

Verification

For x = 60°:

cos 120° = −1/2

For x = 120°:

cos 240° = −1/2

For x = 240°:

cos 480° = −1/2

For x = 300°:

cos 600° = −1/2

All four solutions satisfy the original equation.

Solving Double-Angle Tangent Equations

Worked Example

Solve:

tan 2x = 1

for:

0° ≤ x < 360°

Tangent equals 1 at:

45° + 180°n

Therefore:

2x = 45° + 180°n

Divide by 2:

x = 22.5° + 90°n

Choose integer values of n within the interval.

For n = 0, 1, 2, 3:

x = 22.5°, 112.5°, 202.5°, 292.5°

Important Observation

The period of tan 2x is:

180°/2 = 90°

This explains why the solutions repeat every 90°.

Solving Double-Angle Equations Using Identities

Sometimes direct angle solving is not the best approach.

Consider an equation containing both sin 2x and sin x.

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

Apply the zero-product property:

sin x = 0

or:

2cos x − 1 = 0

First Case

sin x = 0

x = 0°, 180°

Second Case

cos x = 1/2

x = 60°, 300°

Final Answer

x = 0°, 60°, 180°, 300°

Why Factorisation Matters

Dividing both sides by sin x would remove the solutions where sin x = 0.

Factoring preserves all possible solutions.

Solving Equations with Cosine Double-Angle Identities

Worked Example

Solve:

cos 2x = sin x

for:

0° ≤ x < 360°

Use:

cos 2x = 1 − 2sin² x

Substitute:

1 − 2sin² x = sin x

Rearrange:

2sin² x + sin x − 1 = 0

Factor:

(2sin x − 1)(sin x + 1) = 0

Therefore:

sin x = 1/2

or:

sin x = −1

First Case

sin x = 1/2

x = 30°, 150°

Second Case

sin x = −1

x = 270°

Final Answer

x = 30°, 150°, 270°

Verification

For x = 30°:

cos 60° = 1/2 = sin 30°

For x = 150°:

cos 300° = 1/2 = sin 150°

For x = 270°:

cos 540° = −1 = sin 270°

All three solutions are valid.

Solving Quadratic Double-Angle Equations

Worked Example

Solve:

2cos 2x − 3cos x + 1 = 0

for:

0° ≤ x < 360°

Use:

cos 2x = 2cos² x − 1

Substitute:

2(2cos² x − 1) − 3cos x + 1 = 0

Expand:

4cos² x − 2 − 3cos x + 1 = 0

Simplify:

4cos² x − 3cos x − 1 = 0

Factor:

(4cos x + 1)(cos x − 1) = 0

Therefore:

cos x = −1/4

or:

cos x = 1

First Case

cos x = −1/4

The reference angle is:

arccos(1/4) ≈ 75.52°

Cosine is negative in Quadrants II and III.

Therefore:

x ≈ 104.48°, 255.52°

Second Case

cos x = 1

x = 0°

Final Answer

x ≈ 0°, 104.48°, 255.52°

The exact non-special-angle solutions may be expressed as arccos(−1/4) and 360° − arccos(−1/4).

Reviewing Half-Angle Identities

Half-angle identities relate trigonometric functions of x/2 to functions of x.

Sine Half-Angle

sin(x/2) = ±√[(1 − cos x)/2]

Cosine Half-Angle

cos(x/2) = ±√[(1 + cos x)/2]

Tangent Half-Angle

tan(x/2) = sin x/(1 + cos x)

Another form is:

tan(x/2) = (1 − cos x)/sin x

These tangent formulas have different denominator restrictions.

Important Sign Rule

The ± sign in the square-root formulas depends on the quadrant containing x/2, not necessarily the quadrant containing x.

When solving half-angle equations, direct angle solving is often simpler than using square-root identities.

Solving Half-Angle Equations Directly

Worked Example

Solve:

sin(x/2) = 1/2

for:

0° ≤ x < 720°

Let:

u = x/2

Then:

0° ≤ u < 360°

Solve:

sin u = 1/2

Therefore:

u = 30°, 150°

Since u = x/2:

x = 60°, 300°

Final Answer

x = 60°, 300°

General Solution

The general solutions are:

x/2 = 30° + 360°n

or:

x/2 = 150° + 360°n

Multiply by 2:

x = 60° + 720°n

or:

x = 300° + 720°n

where n ∈ ℤ.

Important Observation

The period of sin(x/2) is 720°, not 360°.

Therefore, the equation has only one full sine cycle over the interval 0° ≤ x < 720°.

Solving Half-Angle Equations over One Revolution

Worked Example

Solve:

cos(x/2) = √2/2

for:

0° ≤ x < 360°

Let:

u = x/2

Then:

0° ≤ u < 180°

Cosine equals √2/2 at:

u = 45°

within this interval.

Therefore:

x/2 = 45°

x = 90°

Why Is There Only One Solution?

The usual cosine solutions in one full revolution are:

45° and 315°.

However, the transformed angle u is restricted to:

0° ≤ u < 180°

Therefore, 315° is outside the permitted interval.

This illustrates why the transformed interval must be considered carefully.

Solving Half-Angle Tangent Equations

Worked Example

Solve:

tan(x/2) = −1

for:

0° ≤ x < 720°

Let:

u = x/2

Then:

0° ≤ u < 360°

Tangent equals −1 at:

u = 135°, 315°

Multiply by 2:

x = 270°, 630°

General Solution

Since tangent repeats every 180°:

x/2 = 135° + 180°n

Therefore:

x = 270° + 360°n

where n ∈ ℤ.

The period of tan(x/2) is 360°.

Solving Equations Using Half-Angle Identities

Sometimes an equation contains both a half-angle expression and a full-angle expression.

Worked Example

Solve:

2sin²(x/2) = 1/2

for:

0° ≤ x < 360°

Use:

2sin²(x/2) = 1 − cos x

Substitute:

1 − cos x = 1/2

Rearrange:

cos x = 1/2

Therefore:

x = 60°, 300°

Verification

For x = 60°:

2sin² 30° = 2(1/2)² = 1/2

For x = 300°:

2sin² 150° = 2(1/2)² = 1/2

Both solutions are valid.

Solving Equations Involving Half-Angle and Full-Angle Terms

Worked Example

Solve:

sin(x/2) = cos x

for:

0° ≤ x < 360°

Let:

u = x/2

Then:

x = 2u

and:

0° ≤ u < 180°

The equation becomes:

sin u = cos 2u

Use:

cos 2u = 1 − 2sin² u

Substitute:

sin u = 1 − 2sin² u

Rearrange:

2sin² u + sin u − 1 = 0

Factor:

(2sin u − 1)(sin u + 1) = 0

Therefore:

sin u = 1/2

or:

sin u = −1

First Case

sin u = 1/2

Within 0° ≤ u < 180°:

u = 30°, 150°

Therefore:

x = 60°, 300°

Second Case

sin u = −1

There are no solutions within 0° ≤ u < 180°.

Final Answer

x = 60°, 300°

Reviewing Triple-Angle Identities

Triple-angle identities express trigonometric functions of 3x using functions of x.

Sine

sin 3x = 3sin x − 4sin³ x

Cosine

cos 3x = 4cos³ x − 3cos x

Tangent

tan 3x = (3tan x − tan³ x)/(1 − 3tan² x)

The tangent quotient form requires tan x to be defined and 1 − 3tan² x ≠ 0.

Triple-angle identities are useful when an equation contains both x and 3x.

Solving Triple-Angle Equations Directly

Worked Example

Solve:

sin 3x = 1/2

for:

0° ≤ x < 360°

Let:

u = 3x

Then:

0° ≤ u < 1080°

Sine equals 1/2 at:

u = 30°, 150°, 390°, 510°, 750°, 870°

Divide each angle by 3:

x = 10°, 50°, 130°, 170°, 250°, 290°

Final Answer

x = 10°, 50°, 130°, 170°, 250°, 290°

General Solution

x = 10° + 120°n

or:

x = 50° + 120°n

where n ∈ ℤ.

Solving Triple-Angle Cosine Equations

Worked Example

Solve:

cos 3x = −1

for:

0° ≤ x < 360°

Cosine equals −1 at:

180° + 360°n

Therefore:

3x = 180° + 360°n

Divide by 3:

x = 60° + 120°n

Within the interval:

x = 60°, 180°, 300°

Solving Triple-Angle Tangent Equations

Worked Example

Solve:

tan 3x = √3

for:

0° ≤ x < 360°

Tangent equals √3 at:

60° + 180°n

Therefore:

3x = 60° + 180°n

Divide by 3:

x = 20° + 60°n

Within the interval:

x = 20°, 80°, 140°, 200°, 260°, 320°

The solutions repeat every 60° because tan 3x has a period of:

180°/3 = 60°.

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

sin x = ±√2/2

Therefore:

x = 45°, 135°, 225°, 315°

Final Answer

x = 0°, 45°, 135°, 180°, 225°, 315°

Solving Triple-Angle Cosine Equations Using Factorisation

Worked Example

Solve:

cos 3x = cos x

for:

0° ≤ x < 360°

Use:

cos 3x = 4cos³ x − 3cos x

Substitute:

4cos³ x − 3cos x = cos x

Rearrange:

4cos³ x − 4cos x = 0

Factor:

4cos x(cos² x − 1) = 0

Therefore:

cos x = 0

or:

cos² x = 1

First Case

cos x = 0

x = 90°, 270°

Second Case

cos x = ±1

x = 0°, 180°

Final Answer

x = 0°, 90°, 180°, 270°

Combining Double-Angle and Triple-Angle Identities

More advanced equations may contain both 2x and 3x.

Worked Example

Solve:

sin 3x = sin 2x

for:

0° ≤ x < 360°

A useful strategy is to move everything to one side:

sin 3x − sin 2x = 0

Use the sum-to-product identity:

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

Substitute:

sin 3x − sin 2x

= 2cos(5x/2)sin(x/2)

Therefore:

2cos(5x/2)sin(x/2) = 0

Apply the zero-product property:

cos(5x/2) = 0

or:

sin(x/2) = 0

First Case

cos(5x/2) = 0

Cosine equals zero at:

5x/2 = 90° + 180°n

Multiply by 2:

5x = 180° + 360°n

Divide by 5:

x = 36° + 72°n

Within the interval:

x = 36°, 108°, 180°, 252°, 324°

Second Case

sin(x/2) = 0

x/2 = 180°n

x = 360°n

Within the interval:

x = 0°

Final Answer

x = 0°, 36°, 108°, 180°, 252°, 324°

Why Sum-to-Product Is Useful

Expanding sin 3x and sin 2x would produce a more complicated polynomial expression.

The sum-to-product identity immediately creates factors that can be solved separately.

This is an example of selecting an efficient strategy rather than expanding every identity automatically.

Solving Equations with Multiple Trigonometric Identities

Worked Example

Solve:

sin 3x − sin x = 0

for:

0° ≤ x < 360°

Use the sum-to-product identity:

sin 3x − sin x

= 2cos 2x sin x

Therefore:

2cos 2x sin x = 0

So:

cos 2x = 0

or:

sin x = 0

First Case

cos 2x = 0

2x = 90° + 180°n

x = 45° + 90°n

Within the interval:

x = 45°, 135°, 225°, 315°

Second Case

sin x = 0

x = 0°, 180°

Final Answer

x = 0°, 45°, 135°, 180°, 225°, 315°

Alternative Method

We could also use:

sin 3x = 3sin x − 4sin³ x

Then:

3sin x − 4sin³ x = sin x

2sin x − 4sin³ x = 0

2sin x(1 − 2sin² x) = 0

This gives the same solutions.

The sum-to-product approach is shorter, while the triple-angle approach provides useful practice with polynomial factorisation.

Solving Equations in Radians

The same methods work when angles are measured in radians.

Worked Example

Solve:

sin 2x = √2/2

for:

0 ≤ x < 2π

Let:

u = 2x

Then:

0 ≤ u < 4π

Sine equals √2/2 at:

u = π/4, 3π/4, 9π/4, 11π/4

Divide by 2:

x = π/8, 3π/8, 9π/8, 11π/8

General Solution

x = π/8 + πn

or:

x = 3π/8 + πn

where n ∈ ℤ.

Solving Half-Angle Equations in Radians

Worked Example

Solve:

cos(x/2) = −1/2

for:

0 ≤ x < 4π

Let:

u = x/2

Then:

0 ≤ u < 2π

Cosine equals −1/2 at:

u = 2π/3, 4π/3

Multiply by 2:

x = 4π/3, 8π/3

Both values lie within the specified interval.

Solving Equations Involving Phase Shifts and Multiple Angles

Worked Example

Solve:

sin(2x + 30°) = 1/2

for:

0° ≤ x < 360°

The general sine solutions are:

2x + 30° = 30° + 360°n

or:

2x + 30° = 150° + 360°n

First Family

2x = 360°n

x = 180°n

Within the interval:

x = 0°, 180°

Second Family

2x = 120° + 360°n

x = 60° + 180°n

Within the interval:

x = 60°, 240°

Final Answer

x = 0°, 60°, 180°, 240°

Verifying Solutions

Verification means substituting proposed solutions into the original equation.

It is especially important after:

• Squaring both sides.

• Dividing by an expression containing x.

• Using substitutions.

• Applying several identities.

• Solving equations with domain restrictions.

Worked Example

Verify:

cos 2x = sin x

for the proposed solutions:

x = 30°, 150°, 270°

Check x = 30°

Left-hand side:

cos 60° = 1/2

Right-hand side:

sin 30° = 1/2

Valid.

Check x = 150°

Left-hand side:

cos 300° = 1/2

Right-hand side:

sin 150° = 1/2

Valid.

Check x = 270°

Left-hand side:

cos 540° = −1

Right-hand side:

sin 270° = −1

Valid.

All three solutions satisfy the original equation.

Avoiding Extraneous Solutions

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

Squaring both sides can introduce extraneous solutions.

Worked Example

Solve:

sin x = √(1 − cos x)

for:

0° ≤ x < 360°

The square root is nonnegative, so any solution must satisfy:

sin x ≥ 0

Square both sides:

sin² x = 1 − cos x

Use:

sin² x = 1 − cos² x

Substitute:

1 − cos² x = 1 − cos x

Simplify:

cos² x = cos x

Factor:

cos x(cos x − 1) = 0

Therefore:

cos x = 0

or:

cos x = 1

The candidates are:

x = 0°, 90°, 270°

Verify Each Candidate

For x = 0°:

sin 0° = 0

√(1 − cos 0°) = 0

Valid.

For x = 90°:

sin 90° = 1

√(1 − cos 90°) = 1

Valid.

For x = 270°:

sin 270° = −1

√(1 − cos 270°) = 1

Not valid.

Final Answer

x = 0°, 90°

The value 270° is extraneous because squaring removed the original sign restriction.

Interpreting Multiple Solutions Graphically

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

For example:

sin 2x = 1/2

represents intersections between:

y = sin 2x

and:

y = 1/2

Over 0° ≤ x < 360°, the sine function completes two cycles and intersects the horizontal line four times.

Why Graphs Are Helpful

Graphs can help identify:

• How many solutions should occur.

• Whether solutions repeat periodically.

• Whether a solution lies within the required interval.

• Whether a proposed answer is reasonable.

• Whether a transformation has introduced an invalid solution.

Graphs are useful for checking results, but exact algebraic methods should be used when exact solutions are required.

Applications in Physics: Harmonic Motion

Multiple-angle equations appear in systems involving oscillations and periodic motion.

Suppose the displacement of an object is:

s(t) = 0.40sin(4πt)

where s is measured in metres and t in seconds.

Find every time during:

0 ≤ t < 1

when:

s = 0.20 m

Step 1: Form the Equation

0.40sin(4πt) = 0.20

Divide:

sin(4πt) = 1/2

Step 2: Determine the Angle Interval

Since:

0 ≤ t < 1

then:

0 ≤ 4πt < 4π

The argument completes two cycles.

Step 3: Solve

The solutions are:

4πt = π/6, 5π/6, 13π/6, 17π/6

Divide by 4π:

t = 1/24, 5/24, 13/24, 17/24 seconds

Approximately:

t ≈ 0.0417, 0.2083, 0.5417, 0.7083 seconds

The object reaches the specified displacement four times during the one-second interval.

Applications in Circular Motion

Suppose a point moves around a circle of radius 6 m with vertical position:

y = 6sin 2θ

Find the angles between 0° and 360° for which:

y = 3 m

Step 1: Form the Equation

6sin 2θ = 3

Divide:

sin 2θ = 1/2

Step 2: Solve

The solutions are:

θ = 15°, 75°, 195°, 255°

Interpretation

The expression sin 2θ completes two cycles as θ increases from 0° to 360°.

Therefore, the specified height occurs four times over the interval.

Applications in Wave Interference

When two waves are combined, their resulting displacement may involve multiple-angle expressions.

Suppose:

y = sin 3x − sin x

Use the sum-to-product identity:

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

Then:

y = 2cos 2x sin x

To find positions where the resultant displacement is zero:

2cos 2x sin x = 0

Therefore:

cos 2x = 0

or:

sin x = 0

For:

0° ≤ x < 360°

the solutions are:

x = 0°, 45°, 135°, 180°, 225°, 315°

These positions correspond to points where the combined wave displacement is zero.

Choosing the Most Efficient Method

Not every multiple-angle equation should be solved by expanding identities.

Consider the following approaches.

Important Principle

Choose the identity that reduces complexity rather than automatically expanding every trigonometric expression.

A Systematic Method for Solving Multiple-Angle Equations

1. Identify the interval and angle units.

2. Determine whether the equation contains a double, triple, or half angle.

3. Decide whether direct angle solving or identity substitution is more efficient.

4. If solving directly, determine the full interval of the transformed angle.

5. If using identities, rewrite the equation in a simpler form.

6. Rearrange the equation so one side equals zero when factorisation is useful.

7. Factor carefully without dividing by expressions that may equal zero.

8. Solve every resulting trigonometric equation.

9. Write general solutions when required.

10. Apply the specified interval.

11. Remove repeated solutions.

12. Check the domain of the original equation.

13. Verify every candidate in the original equation.

14. State the final answer clearly.

Common Mistakes

Failing to adjust the angle interval

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

Using the wrong period

sin 2x has a period of 180°, while sin(x/2) has a period of 720°.

Confusing sin 2x with sin² x

These expressions are not equivalent.

Confusing sin 3x with sin³ x

sin 3x means sin(3x), not the cube of sin x.

Using the wrong double-angle identity

Choose the form that matches the functions already present in the equation.

Forgetting both roots of a quadratic

Factorisation may produce more than one trigonometric equation.

Dividing by a trigonometric expression that may equal zero

This can remove valid solutions.

Ignoring the sign in half-angle identities

The sign depends on the quadrant of the half angle.

Assuming half-angle equations have the same number of solutions as ordinary equations

Half-angle functions have longer periods.

Using 360° as the period of tangent

Tangent repeats every 180°.

Failing to remove duplicate solutions

Different factors may produce the same angle.

Including solutions outside the interval

Always check the interval after solving.

Ignoring domain restrictions

Tangent and rational trigonometric expressions may be undefined at certain angles.

Squaring without verifying

Squaring may introduce extraneous solutions.

Rounding too early

Keep exact values whenever possible.

Failing to verify answers

Every candidate should satisfy the original equation.

Key Terms

Multiple-Angle Equation: A trigonometric equation involving a multiple or fraction of an unknown angle.

Double-Angle Equation: An equation containing a trigonometric function of twice an angle.

Triple-Angle Equation: An equation containing a trigonometric function of three times an angle.

Half-Angle Equation: An equation containing a trigonometric function of half an angle.

Double-Angle Identity: A formula expressing a trigonometric function of 2x using functions of x.

Triple-Angle Identity: A formula expressing a trigonometric function of 3x using functions of x.

Half-Angle Identity: A formula expressing a trigonometric function of x/2 using functions of x.

Compound-Angle Identity: A formula 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.

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

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

General Solution: A formula representing every solution to an equation using an integer parameter.

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

Transformed Angle: A new angle expression introduced to simplify an equation.

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

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

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.

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.

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

Identity Substitution: Replacing a trigonometric expression with an equivalent expression.

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

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

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

Phase Shift: A horizontal displacement of a periodic function.

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

Frequency: The number of complete cycles per unit time.

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

Oscillation: Repeated motion around an equilibrium position.

Harmonic Motion: Periodic motion that can be described using sinusoidal functions.

Wave Interference: The combination of two or more waves to produce a resultant wave.

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

Check Your Understanding

1. Explain how multiplying an angle by 2 or 3 changes the period of a sine function.

2. State the periods of sin 2x, cos 3x, tan 2x, and sin(x/2) in degrees.

3. Solve:

sin 2x = √3/2

for 0° ≤ x < 360°.

4. Solve:

cos 2x = −√2/2

for 0° ≤ x < 360°.

5. Solve:

tan 3x = 1

for 0° ≤ x < 360°.

6. Solve:

sin(x/2) = √2/2

for 0° ≤ x < 720°.

7. Solve:

cos(x/2) = −1/2

for 0° ≤ x < 720°.

8. Solve:

sin 2x = sin x

for 0° ≤ x < 360°.

9. Solve:

cos 2x = cos x

for 0° ≤ x < 360°.

10. Solve:

2sin²(x/2) = 3/2

for 0° ≤ x < 360°.

11. Solve:

sin 3x = sin x

for 0° ≤ x < 360°.

12. Solve:

cos 3x = cos x

for 0° ≤ x < 360°.

13. Solve:

sin 3x = sin 2x

for 0° ≤ x < 360°. Use a sum-to-product identity.

14. Solve:

cos(2x + 30°) = 1/2

for 0° ≤ x < 360°. Verify every solution.

15. A vibrating object has displacement:

s(t) = 0.25sin(6πt)

where s is measured in metres and t in seconds.

Find every time during 0 ≤ t < 1 when the displacement equals 0.125 m. Explain how the angular frequency affects the number of solutions, and verify your answers.

Key Takeaways

• Multiple-angle equations contain trigonometric functions of double, triple, or half angles.

• Multiplying the angle decreases the period of a trigonometric function.

• Dividing the angle increases the period.

• Double-angle equations may be solved directly or by applying double-angle identities.

• Triple-angle equations may be solved directly or by using triple-angle identities.

• Half-angle equations require careful attention to the transformed angle interval.

• The sine double-angle identity is sin 2x = 2sin x cos x.

• The cosine double-angle identity has three equivalent forms.

• The sine triple-angle identity is sin 3x = 3sin x − 4sin³ x.

• The cosine triple-angle identity is cos 3x = 4cos³ x − 3cos x.

• Half-angle identities connect functions of x/2 with functions of x.

• Direct angle solving is often the simplest method when only one trigonometric function is present.

• Identity substitution is useful when equations contain several related trigonometric expressions.

• Factorisation allows complicated equations to be separated into simpler cases.

• Sum-to-product identities can provide efficient alternatives to expanding triple-angle expressions.

• The zero-product property is essential for solving factorised trigonometric equations.

• Dividing by a variable trigonometric expression may eliminate valid solutions.

• The transformed angle interval must be considered when finding every solution.

• General solutions describe all possible answers using integer multiples of a period.

• Specified intervals determine which solutions belong in the final answer.

• Squaring equations can introduce extraneous solutions.

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

• Graphs help explain why multiple-angle equations may have several solutions.

• Multiple-angle equations are used in oscillations, circular motion, wave interference, and other periodic systems.

• Mastering multiple-angle equations develops the trigonometric fluency, algebraic reasoning, and verification skills needed for advanced mathematics, calculus, and physics.