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.