Compound angle formulae

The compound angle formulae expand the sine, cosine or tangent of a sum or difference of two angles: sin(A ± B) = sin A cos B ± cos A sin B, cos(A ± B) = cos A cos B ∓ sin A sin B and tan(A ± B) = (tan A ± tan B) ÷ (1 ∓ tan A tan B). They let you find exact values such as sin 75°, solve equations like cos(x − 60°) = sin x, and write a cos θ + b sin θ as a single wave, R cos(θ − α). They are also called the addition formulae, or the sum and difference identities.

The compound angle formulae

Two patterns help you remember them:

• Sine keeps the sign and mixes the functions: sin, cos, then cos, sin, with the same sign as in the bracket.
• Cosine flips the sign and keeps the functions together: cos, cos, then sin, sin, with the opposite sign to the bracket.

The tan formulae fail when the bottom is zero, which happens when A ± B is 90°, 270° and so on, where tan is undefined.

A common wrong guess

sin(A + B) is not sin A + sin B. Try A = B = 30°: sin 60° = 0.866, but sin 30° + sin 30° = 1. The function does not split across a bracket like multiplication does.

Proof of the compound angle formula for sin(A + B)

The altitude h splits the triangle into two right-angled triangles. Use area = ½ab sin C on each part and on the whole:

• Left part: ½ × p × h × sin A
• Right part: ½ × q × h × sin B
• Whole triangle: ½ × p × q × sin(A + B)

The whole is the sum of the parts. From the right-angled triangles, h = q cos B in the left area and h = p cos A in the right area, so:

Dividing by ½pq gives sin(A + B) = sin A cos B + cos A sin B. Replacing B with −B, and using sin(−B) = −sin B and cos(−B) = cos B, gives sin(A − B). The cosine formulae follow from cos θ = sin(90° − θ), and the tan formulae from dividing the sin formula by the cos formula.

Exact values with the compound angle formula

Write the angle as a sum or difference of 30°, 45° and 60°, whose exact values you know: sin 30° = ½, cos 30° = √3/2, sin 45° = cos 45° = √2/2, tan 45° = 1, tan 60° = √3.

Example 1: sin 75°

Check on a calculator: both sides are 0.9659.

Example 2: cos 105°

The answer is negative, as it should be for an angle between 90° and 180°.

Example 3: tan 105°

Rationalise the denominator by multiplying top and bottom by 1 + √3: the top becomes (√3 + 1)² = 4 + 2√3 and the bottom 1 − 3 = −2, so tan 105° = −2 − √3.

Example 4: from given ratios

A and B are acute angles with sin A = 3/5 and cos B = 5/13. Find the exact value of sin(A + B).

First find the missing ratios with Pythagoras: cos A = 4/5, from the 3, 4, 5 triangle, and sin B = 12/13, from the 5, 12, 13 triangle.

Now try a question in the course. Choose the simplified form, then press Mark to check it there.

Solving equations with compound angles

Example 5: cos(x − 60°) = sin x

Solve for 0° ≤ x < 360°. Expand the left side:

Collect the sin x terms: ½ cos x = (1 − √3/2) sin x. Divide by cos x to get tan x:

Since tan 75° = 2 + √3 (it is 1 ÷ tan 15°), x = 75° or 255°, adding 180° for the second solution because tan repeats every 180°.

The R cos(θ − α) form

Any expression a cos θ + b sin θ is a single cosine wave in disguise. Writing it as R cos(θ − α) shows its maximum, its minimum and where they happen, and turns hard equations into easy ones. It is sometimes called the harmonic form.

Expand the target with the compound angle formula: R cos(θ − α) = R cos θ cos α + R sin θ sin α. Matching the coefficients with a cos θ + b sin θ gives:

Squaring and adding gives R = √(a² + b²), because cos²α + sin²α = 1. Dividing gives tan α = b ÷ a.

Example 6: 5 cos θ + 12 sin θ

Write 5 cos θ + 12 sin θ in the form R cos(θ − α), with R > 0 and 0° < α < 90°.

So 5 cos θ + 12 sin θ = 13 cos(θ − 67.4°). Its maximum value is 13, when θ = 67.4°, and its minimum is −13, when θ = 247.4°.

Here x is the angle in radians and y is the value of 5 cos x + 12 sin x. The curve is an ordinary cosine wave stretched to height 13 and shifted right by α = 1.176 rad, which is 67.4°.

Example 7: solving with the R form

Solve 5 cos θ + 12 sin θ = 6.5 for 0° ≤ θ < 360°.

Using Example 6, 13 cos(θ − 67.4°) = 6.5, so cos(θ − 67.4°) = 0.5. Then θ − 67.4° = 60° or −60° (both in range once α is added), giving:

The same method works for R sin(θ ± α) and R cos(θ + α): expand the form you want and match coefficients.

Where else the compound angle formulae appear

• Double angle formulae: put B = A in sin(A + B) to get sin 2A = 2 sin A cos A, and similarly for cos 2A and tan 2A. Our guide to the double angle formula covers these.
• Differentiating sin x from first principles: the expansion of sin(x + h) is the first step.
• Waves in physics: adding two waves of the same frequency gives another wave of that frequency, which is the R form in action.

The full list of identities, including the reciprocal and Pythagorean ones, is in our trig identities guide.

Common mistakes

• Splitting the bracket: sin(A + B) ≠ sin A + sin B.
• The sign in cos(A + B). It is a minus: cos A cos B − sin A sin B.
• The signs in tan(A − B). Minus on top, plus underneath.
• Wrong quadrant for α. With a and b both positive, α is acute. Check that R cos α and R sin α have the signs of a and b.
• Losing solutions. After solving cos(θ − α) = k, find every value of θ − α in the shifted range before adding α.

Is the compound angle formula in the formula booklet?

Yes, for AQA. AQA's AS and A-level Mathematics formulae booklet, checked in November 2026, gives sin(A ± B), cos(A ± B) and tan(A ± B) under trigonometric identities, with the condition that A ± B is not an odd multiple of 90° for tan. The double angle formulae are not given, so you need to derive or learn them. Other boards publish their own booklets, so check yours.

Practise trigonometry in a short timed paper

A-level Maths: Algebra & Functions gives you a short timed paper every day, with 8 questions worth 25 to 35 marks across the course, so trig identities turn up regularly but not in every paper. The Trigonometry course builds up from right-angled triangles to identities.

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Maths · TrigonometryHarmonic form: a sin x + b cos x as R sin(x + α)How to write a sin x + b cos x in harmonic form R sin(x ± α) or R cos(x ± α): finding R and α, maximum and minimum values, and solving equations.