Compound angle formulae

The formulae

• sin(A ± B) ≡ sin A cos B ± cos A sin B
• cos(A ± B) ≡ cos A cos B ∓ sin A sin B (note the sign swaps)
• tan(A ± B) ≡ (tan A ± tan B)/(1 ∓ tan A tan B)

sin(A + B) is not sin A + sin B.

Exact values

Write the angle as a sum or difference of 30°, 45° and 60°:

cos 75° = cos(45° + 30°) = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2) = (sqrt(6) - sqrt(2))/4.

Equations and given ratios

Expand, then collect sin x and cos x terms and divide to get tan x. If you are given sin A = 3/5 for an acute angle, draw a 3, 4, 5 triangle to get cos A = 4/5.

Worked example

Find the exact value of sin 15°.

1. sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30°
2. = (sqrt(2)/2)(sqrt(3)/2) - (sqrt(2)/2)(1/2)

Answer: (sqrt(6) - sqrt(2))/4

Check yourself

1. Find the exact value of cos 75°.
   • (sqrt(6) + sqrt(2))/4
   • (sqrt(6) - sqrt(2))/4
   • (sqrt(3) - 1)/2
   • (sqrt(2) - sqrt(6))/4
   Show answer

   (sqrt(6) - sqrt(2))/4. cos(45° + 30°) = cos 45° cos 30° - sin 45° sin 30° = sqrt(6)/4 - sqrt(2)/4.

2. A and B are acute, sin A = 3/5 and cos B = 5/13. Find sin(A + B). Give your answer as a fraction in its simplest form.
   Show answer

   63/65. cos A = 4/5 and sin B = 12/13. sin(A + B) = (3/5)(5/13) + (4/5)(12/13) = 15/65 + 48/65 = 63/65.

3. tan A = 1/2 and tan B = 1/3. Find tan(A + B).
   Show answer

   1. (1/2 + 1/3)/(1 - 1/6) = (5/6)/(5/6) = 1.

4. Which is the expansion of cos(A - B)?
   • cos A cos B - sin A sin B
   • cos A - cos B
   • cos A cos B + sin A sin B
   • sin A cos B - cos A sin B
   Show answer

   cos A cos B + sin A sin B. For cos, the sign in the middle is the opposite of the sign in the bracket.

5. Solve sin(x + 30°) = 2cos x for 0° < x < 180°.
   Show answer

   60 °. sin x cos 30° + cos x sin 30° = 2cos x gives (sqrt(3)/2)sin x = (3/2)cos x, so tan x = sqrt(3) and x = 60°.

6. Find the exact value of tan 15°.
   • sqrt(3) - 1
   • 2 + sqrt(3)
   • 2 - sqrt(3)
   • (sqrt(3) - 1)/2
   Show answer

   2 - sqrt(3). tan(45° - 30°) = (1 - 1/sqrt(3))/(1 + 1/sqrt(3)) = (sqrt(3) - 1)/(sqrt(3) + 1) = (4 - 2sqrt(3))/2 = 2 - sqrt(3).

7. Simplify sin(x + 90°).
   • cos x
   • sin x
   • -cos x
   • -sin x
   Show answer

   cos x. sin x cos 90° + cos x sin 90° = sin x(0) + cos x(1) = cos x.

Practise with instant feedback Print this lesson

Sources

• Pearson Edexcel International Advanced Level in Mathematics specification (issue 3)
• Pearson Edexcel IAL Pure Mathematics 3 scheme of work

Last updated 7 November 2026. Spotted a mistake? Tell us.