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Compound angle formulae

Derive and use sine, cosine and tangent addition formulae. Explore signed components, expand compound angles and practise with worked solutions.

Before you startUnit-circle coordinates, exact trig values and basic algebra

01 / Add angles, not trig values

A second rotation changes both coordinates.

For a unit vector at angle A, its coordinates are (cos A, sin A). Turn it through another angle B and its coordinates become (cos(A + B), sin(A + B)). The new coordinates are not obtained by adding the two original sine or cosine values.

sin(A + B) is generally different from sin A + sin B

Choose A and B yourself. The blue segment follows the A direction; the green segment is perpendicular to it. Follow them head to tail to reach the gold endpoint. The written derivation in the next sections stays available while you investigate.

Add two signed componentsExplore

Blue: cos B along the A direction. Green: sin B along the direction 90° anticlockwise from A. Gold: their sum. Negative B turns clockwise. Decimals are rounded to 4 places.

A + B = 90°. x = 0.4330 − 0.4330 = 0.0000; y = 0.2500 + 0.7500 = 1.0000.

02 / Build the rotated vector

Resolve it along two perpendicular directions.

A unit vector in the A direction has coordinates u = (cos A, sin A). A unit vector 90° anticlockwise from it has coordinates v = (−sin A, cos A). These two directions form perpendicular axes turned through A.

Vector at angle A + B = (cos B)u + (sin B)v

Relative to these turned axes, the vector makes angle B, so its components are cos B and sin B. Multiplying out the two coordinate pairs gives:

x = cos A cos B − sin A sin B
y = sin A cos B + cos A sin B

The components are signed. A negative coefficient reverses a direction, so this argument also works outside acute triangles. When B = −30°, the second component points clockwise from u.

Watch two components rotate together

Pause, replay or seek freely. The notes explain the same idea and stay in view.

03 / Read the addition formulae

Match the endpoint coordinates in two ways.

The same endpoint has x-coordinate cos(A + B) and y-coordinate sin(A + B). Equating these with the component expressions proves the addition formulae for all real angles:

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

Check A = 30°, B = 60°Worked example

sin 90° = (1/2)(1/2) + (√3/2)(√3/2)

The right side is 1/4 + 3/4 = 1.

cos 90° = (√3/2)(1/2) − (1/2)(√3/2)

The two horizontal contributions cancel.

The sine addition formula mixes one sine and one cosine in each term. The cosine addition formula pairs like functions and subtracts the sine product.

04 / Turn backwards to subtract an angle

Use the parity of sine and cosine.

Replace B by −B in the addition formulae. Cosine is even, so cos(−B) = cos B; sine is odd, so sin(−B) = −sin B.

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

Expand cos(2x − π/6)Worked example

Treat 2x as A and π/6 as B

The angle need not be a single letter.

cos 2x cos(π/6) + sin 2x sin(π/6)

The difference inside cosine gives a plus between products.

(√3/2)cos 2x + (1/2)sin 2x

This holds for every real x.

05 / Expand an expression systematically

Keep coefficients outside until the brackets are correct.

Simplify 2sin(x + π/3) − cos xWorked example

2[sin x cos(π/3) + cos x sin(π/3)] − cos x

Expand the compound angle first.

sin x + √3 cos x − cos x

Multiply both terms by 2.

sin x + (√3 − 1)cos x

Collect like trig functions.

Simplify cos(x − π/4) − sin(x − π/4)Worked example

(cos x + sin x)/√2 − (sin x − cos x)/√2

Keep the subtraction outside the second bracket.

√2 cos x

The sine terms cancel; the cosine terms add.

06 / Recognise a compound angle in reverse

Identify both angles and the sign before calculating.

Evaluate sin 70° cos 20° − cos 70° sin 20°Worked example

This is sin(A − B), with A = 70°, B = 20°

The products are mixed and separated by a minus.

The expression is sin 50°

Combining is exact even when the final angle has no simple surd.

Evaluate cos(π/8)cos(3π/8) − sin(π/8)sin(3π/8)Worked example

This is cos(π/8 + 3π/8)

Like-function products with a minus.

cos(π/2) = 0

No individual eighth-angle values are needed.

Order matters for a sine difference: swapping A and B changes the sign. A cosine difference does not change, because cosine is even.

07 / Derive tangent from sine and cosine

Divide only where each denominator is nonzero.

Start from tan(A + B) = sin(A + B)/cos(A + B). Substitute the addition formulae, then divide every numerator and denominator term by cos A cos B.

tan(A + B) = (tan A + tan B)/(1 − tan A tan B)
tan(A − B) = (tan A − tan B)/(1 + tan A tan B)

These displayed identities apply where tan A and tan B exist and the displayed denominator is nonzero. They work in degrees or radians, provided both angles use the same unit.

Given tan A = 1/2 and tan B = 1/3, find tan(A + B)Worked example

(1/2 + 1/3)/(1 − 1/6)

Both tangents are supplied and the denominator is 5/6.

(5/6)/(5/6) = 1

The angle A + B need not be 45° unless an interval is also specified.

08 / Keep track of where a formula applies

A component tangent can be undefined even when the final tangent exists.

Consider A = 90°, B = 45°Worked example

tan(A + B) = tan 135° = −1

The original combined angle is valid.

tan A is undefined

The quotient written using tan A and tan B cannot be used here.

Use sin(A + B)/cos(A + B) instead

The sine/cosine addition formulae give (√2/2)/(−√2/2) = −1.

If tan A tan B = 1 and both tangents exist, the addition quotient has denominator zero. This corresponds to cos(A + B) = 0, so the combined tangent really is undefined. Always distinguish these two situations.

09 / Use expansion to prove a relationship

A proof explains why an equality holds throughout its domain.

Prove sin(x + h) + sin(x − h) = 2sin x cos hWorked example

Expand both compound sines

sin x cos h + cos x sin h + sin x cos h − cos x sin h.

Cancel the opposite terms

The two remaining terms are both sin x cos h.

A student writes cos(A + B) = cos A + cos B. Disprove this using an angle pair different from A = B = 0°.

Hint

Choose angles whose cosine values you know exactly.

Worked solution

For example, A = B = 60°: the left side is cos 120° = −1/2; the right side is 1/2 + 1/2 = 1. One counterexample disproves the identity.

11 / Recap

Track the products, signs and domains.

Sine addition mixes sine and cosine; cosine addition pairs like functions.
For subtraction, replace B by −B and use parity.
Expand compound expressions before collecting like terms.
Recognise the same formulae backwards to combine products.
Tangent quotients require both component tangents and a nonzero denominator.
A geometric component proof uses signed coordinates, so it is not restricted to acute angles.

Section 1 of 11 · Add angles, not trig values