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Learn · Mathematics · Grade 11 · Trigonometry

Sum, difference and double-angle formulas

sin(a ± b), cos(a ± b), tan(a ± b), the double-angle and half-angle formulas, and exact values such as sin 15° and cos 75°.

High school Lesson 27 of 31 Interactive figure Practice

First, the rules

Angles you can add

sin(a + b) is NOT sin a + sin b. These formulas say what it really is — and they unlock 15°, 75°, 105° and every double angle.

★ Sum and difference
Examples 1, 2
sin(a ± b) = sin a cos b ± cos a sin b
cos(a ± b) = cos a cos b ∓ sin a sin b
tan(a ± b) = tan a ± tan b1 ∓ tan a tan b
Sine keeps the sign, cosine flips it.
Memory hook
sin: "sin cos, cos sin" — mixed. cos: "cos cos, sin sin" — matched, with the opposite sign.
★ Double angle
Example 3
sin 2a = 2 sin a cos a
cos 2a = cos²a − sin²a = 1 − 2 sin²a = 2 cos²a − 1
tan 2a = 2 tan a1 − tan²a
Half angle (from cos 2a)
Example 3
sin²a = 1 − cos 2a2, cos²a = 1 + cos 2a2
★ Exact values this unlocks
Example 1
15° = 45° − 30°, 75° = 45° + 30°, 105° = 60° + 45°

See it

Add two angles on the circle

Set a and b. The pink arm is a + b: read sin(a + b) from its height, then compare with sin a cos b + cos a sin b. Make b = a to see the double-angle formulas appear.

Step by step

Worked examples

1

sin 75° and cos 15°

sum formula

Find the exact values of sin 75° and cos 15°.

Ask yourself: Write 75° as 45° + 30°; both pieces have known values.

Show the solution
  1. sin 75° = sin 45° cos 30° + cos 45° sin 30° = √22 · √32 + √22 · 12 = √6 + √24.
  2. cos 15° = cos(45° − 30°) = cos 45° cos 30° + sin 45° sin 30° — the same two products, same sign: also √6 + √24.
  3. That is no accident: cos 15° = sin(90° − 15°) = sin 75°.

Answer both equal √6 + √24 ≈ 0.966

2

Simplifying with the formulas backwards

recognise the pattern

Simplify sin 50° cos 20° − cos 50° sin 20° and cos 35° cos 25° − sin 35° sin 25°.

Ask yourself: Each expression is the right-hand side of a formula. Which one, and with which sign?

Show the solution
  1. sin a cos b − cos a sin b = sin(a − b): sin(50° − 20°) = sin 30° = 12.
  2. cos a cos b − sin a sin b = cos(a + b): cos 60° = 12.

Answer 12 and 12

3

Double angles from one ratio

double angle

sin x = 35 and x is acute. Find sin 2x, cos 2x and tan 2x.

Ask yourself: First get cos x from the 3-4-5 triangle; then every double-angle formula is just arithmetic.

Show the solution
  1. cos x = 45 (3-4-5).
  2. sin 2x = 2 · 35 · 45 = 2425.
  3. cos 2x = 1 − 2 · 925 = 725.
  4. tan 2x = sin 2xcos 2x = 247.

Answer 2425, 725, 247

Practice: sum and double angles

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Exact values of 15°, 75° and 105°, double angles from a given ratio, and recognising a formula written backwards. Your best score is kept in this browser only.

About this lesson

Sum, difference and double-angle formulas: sin(a ± b), cos(a ± b), tan(a ± b), the double-angle formulas for sin 2a, cos 2a and tan 2a, the half-angle forms, and exact values such as sin 15° and cos 75° built from 45° and 30°. Worked examples and practice with exact answers.

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