Compound Angle Formulae — Edexcel A-level Mathematics (Pure)

SPEC 5.6 • TRIGONOMETRY

Addition Formulae

Compound angle identities for sine, cosine and tangent

Learning Objectives

By the end of this lesson you will be able to:

Know and apply the addition formulae for sin(A ± B), cos(A ± B), and tan(A ± B)

Use addition formulae to find exact values of trigonometric functions

Apply addition formulae to simplify expressions

Use addition formulae to prove identities

Why Addition Formulae?

Sometimes we need to find exact values of trigonometric functions at angles that are not "standard" angles like 30°, 45°, or 60°.

Problem

What is the exact value of

sin 75°

?

75° is not a standard angle, but: 75° = 45° + 30°

Common Error

sin (A + B) \neq = sin A + sin B

For example:

sin 45° + sin 30° = \frac{2}{2} + \frac{1}{2} \neq = sin 75°

The sin(A ± B) Formulae

Sine Addition Formulae

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

sin (A - B) = sin A cos B - cos A sin B

Memory Tip

For sine: "Sine Cosine, Cosine Sine" — alternate between sin and cos

The sign in the middle matches the sign in the bracket

The cos(A ± B) Formulae

Cosine Addition Formulae

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

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

Watch Out

The sign in the middle is opposite to the sign in the bracket!

cos(A + B) has a minus sign | cos(A − B) has a plus sign

The tan(A ± B) Formulae

Tangent Addition Formulae

tan (A + B) = \frac{tan A + tan B}{1 - tan A tan B}

tan (A - B) = \frac{tan A - tan B}{1 + tan A tan B}

Pattern

Numerator: tan A ± tan B (sign matches bracket)

Denominator: 1 ∓ tan A tan B (sign is opposite)

Formulae Summary

Sine

sin (A \pm B) = sin A cos B \pm cos A sin B

Cosine

cos (A \pm B) = cos A cos B \mp sin A sin B

Tangent

tan (A \pm B) = \frac{tan A \pm tan B}{1 \mp tan A tan B}

Exam Tip

These formulae are given in the formula booklet. Focus on using them correctly!

Worked Example01

Find the exact value of $sin 75°$ .

1

Express 75° as a sum:

75° = 45° + 30°

2

Apply

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

3

sin 75° = sin 45° cos 30° + cos 45° sin 30°

4

= \frac{2}{2} \times \frac{3}{2} + \frac{2}{2} \times \frac{1}{2} = \frac{6}{4} + \frac{2}{4}

Answer

sin 75° = \frac{6 + 2}{4}

Worked Example02

Find the exact value of $cos 15°$ .

1

Express 15° as a difference:

15° = 45° - 30°

2

Apply

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

3

cos 15° = cos 45° cos 30° + sin 45° sin 30°

4

= \frac{2}{2} \times \frac{3}{2} + \frac{2}{2} \times \frac{1}{2} = \frac{6}{4} + \frac{2}{4}

Answer

cos 15° = \frac{6 + 2}{4}

Worked Example03

Find the exact value of $tan 105°$ .

1

Express 105° as a sum:

105° = 60° + 45°

2

Apply

tan (A + B) = \frac{t a n A + t a n B}{1 - t a n A t a n B}

3

tan 105° = \frac{3 + 1}{1 - 3}

4

Rationalise:

= \frac{( 3 + 1 ) ( 1 + 3 )}{( 1 - 3 ) ( 1 + 3 )} = \frac{4 + 2 3}{- 2}

Answer

tan 105° = - 2 - 3

Worked Example04

Simplify: $sin (x + 30°) + sin (x - 30°)$

1

Expand:

sin (x + 30°) = \frac{3}{2} sin x + \frac{1}{2} cos x

2

Expand:

sin (x - 30°) = \frac{3}{2} sin x - \frac{1}{2} cos x

3

Add:

\frac{3}{2} sin x + \frac{1}{2} cos x + \frac{3}{2} sin x - \frac{1}{2} cos x

Answer

sin (x + 30°) + sin (x - 30°) = 3 sin x

Worked Example05

Simplify: $cos (x + 45°) + cos (x - 45°)$

1

Expand:

cos (x + 45°) = \frac{2}{2} cos x - \frac{2}{2} sin x

2

Expand:

cos (x - 45°) = \frac{2}{2} cos x + \frac{2}{2} sin x

3

Add: The

sin x

terms cancel out

Answer

cos (x + 45°) + cos (x - 45°) = 2 cos x

Worked Example06

Given $sin A = \frac{3}{5}$ (A acute) and $cos B = \frac{12}{13}$ (B acute), find $sin (A + B)$ .

1

Find

cos A

:

cos^{2} A = 1 - \frac{9}{25} = \frac{16}{25}

, so

cos A = \frac{4}{5}

2

Find

sin B

:

sin^{2} B = 1 - \frac{144}{169} = \frac{25}{169}

, so

sin B = \frac{5}{13}

3

Apply:

sin (A + B) = \frac{3}{5} \times \frac{12}{13} + \frac{4}{5} \times \frac{5}{13}

4

= \frac{36}{65} + \frac{20}{65} = \frac{56}{65}

Answer

sin (A + B) = \frac{56}{65}

Worked Example07

Show that $sin (θ + 60°) - sin (θ - 60°) = 3 cos θ$

1

Expand:

sin (θ + 60°) = \frac{1}{2} sin θ + \frac{3}{2} cos θ

2

Expand:

sin (θ - 60°) = \frac{1}{2} sin θ - \frac{3}{2} cos θ

3

Subtract:

(\frac{1}{2} sin θ + \frac{3}{2} cos θ) - (\frac{1}{2} sin θ - \frac{3}{2} cos θ)

4

= \frac{3}{2} cos θ + \frac{3}{2} cos θ = 3 cos θ

✓

P

Practice 1

Find the exact value of $sin 15°$ .

Hint: 15° = 45° − 30°

1

sin 15° = sin (45° - 30°) = sin 45° cos 30° - cos 45° sin 30°

2

= \frac{2}{2} \times \frac{3}{2} - \frac{2}{2} \times \frac{1}{2} = \frac{6}{4} - \frac{2}{4}

Answer

sin 15° = \frac{6 - 2}{4}

P

Practice 2

Simplify: $cos (x + 60°) - cos (x - 60°)$

1

cos (x + 60°) = \frac{1}{2} cos x - \frac{3}{2} sin x

2

cos (x - 60°) = \frac{1}{2} cos x + \frac{3}{2} sin x

3

Subtracting:

- \frac{3}{2} sin x - \frac{3}{2} sin x = - 3 sin x

Answer

cos (x + 60°) - cos (x - 60°) = - 3 sin x

Summary

Key Formulae

sin (A \pm B) = sin A cos B \pm cos A sin B

cos (A \pm B) = cos A cos B \mp sin A sin B

tan (A \pm B) = \frac{t a n A \pm t a n B}{1 \mp t a n A t a n B}

Remember

Common angle splits: 75° = 45° + 30°, 15° = 45° − 30°, 105° = 60° + 45°

What's Next?

In the next lesson, we'll explore Double Angle Formulae — special cases where A = B.