Angle addition formulae for A-Level Maths
The angle addition formulae are six identities that tell you the sine, cosine and tangent of (A + B) and (A - B) in terms of the sines, cosines and tangents of A and B on their own. They sit at the heart of A-Level trigonometry and feed directly into double-angle formulae, R-form expressions, and solving trig equations on the Edexcel papers. In plain language: If you know sin A, cos A, sin B and cos B, you can work out sin(A + B), cos(A + B) and tan(A + B) without going back to the unit circle.
This guide lists the six formulae, shows how to use them, derives the double-angle formulae from them, walks through a worked example, and covers the mistakes that cost the most marks at A-Level.
Six formulae
Three for the sum and three for the difference (sin, cos, tan of A + B and A - B). All six are in the Edexcel formula book.
Lead to double-angle
Set B = A and the addition formulae give you the double-angle identities for sin 2A, cos 2A and tan 2A in a single step.
Core to R-form and proofs
R sin(x + a) form, harmonic identities and many proof-by-deduction questions all start with an angle addition formula.
The six formulae
All six are in the Edexcel A-Level formula book, but you should still know them well enough to use them without flipping back. Speed matters in C3-style trig questions.
| Identity | Formula | Memory hook |
|---|---|---|
| sin(A + B) | sin A cos B + cos A sin B | Sine: Mixed pairs, plus sign matches |
| sin(A - B) | sin A cos B - cos A sin B | Sine: Mixed pairs, minus sign matches |
| cos(A + B) | cos A cos B - sin A sin B | Cosine: Same pairs, sign flips |
| cos(A - B) | cos A cos B + sin A sin B | Cosine: Same pairs, sign flips |
| tan(A + B) | (tan A + tan B) / (1 - tan A tan B) | Tangent: Sum on top, 1 minus product on bottom |
| tan(A - B) | (tan A - tan B) / (1 + tan A tan B) | Tangent: Difference on top, 1 plus product on bottom |
Why the sign flips for cosine The most common error is writing cos(A + B) = cos A cos B + sin A sin B. It is minus, not plus. A quick check: Try A = B = 0. Then cos(A + B) = cos 0 = 1, and cos A cos B - sin A sin B = 1 - 0 = 1. The version with a plus sign would give 1 + 0 = 1 too, so try A = B = 90 degrees. Then cos(180) = -1, and the correct formula gives 0 - 1 = -1.
Using the formulae to find exact values
A standard exam question gives you an angle that is not on the unit circle (like 75 degrees) and asks for an exact value. The trick is to split the awkward angle into two angles you do know exactly.
75 degrees = 45 degrees + 30 degrees, and you know exact values for both. So sin 75 = sin(45 + 30) = sin 45 cos 30 + cos 45 sin 30. Substituting in the surd values from the formula sheet gives (root 6 + root 2) divided by 4.
Worked example: Finding sin 15 in surd form
Question: Find the exact value of sin 15 degrees in surd form.
Step 1: Split 15 into two known angles. 15 = 45 - 30.
Step 2: Apply sin(A - B) = sin A cos B - cos A sin B with A = 45 and B = 30. sin 15 = sin 45 cos 30 - cos 45 sin 30.
Step 3: Substitute exact values. sin 45 = root 2 / 2, cos 30 = root 3 / 2, cos 45 = root 2 / 2, sin 30 = 1 / 2.
Step 4: Multiply through. sin 15 = (root 2 / 2)(root 3 / 2) - (root 2 / 2)(1 / 2) = root 6 / 4 - root 2 / 4.
Step 5: Combine. sin 15 = (root 6 - root 2) / 4.
As a sanity check, sin 15 should be about 0.2588 and (root 6 - root 2) / 4 is roughly (2.449 - 1.414) / 4 = 0.259. The exact answer matches.
Deriving the double-angle formulae
Double-angle formulae are the most useful spin-off from the addition formulae. To get them, set B = A in each addition formula.
For sine: Sin(A + A) = sin A cos A + cos A sin A = 2 sin A cos A. So sin 2A = 2 sin A cos A.
For cosine: Cos(A + A) = cos A cos A - sin A sin A = cos squared A - sin squared A. Using sin squared A + cos squared A = 1, you get three equivalent forms: Cos 2A = cos squared A - sin squared A = 2 cos squared A - 1 = 1 - 2 sin squared A.
For tangent: Tan 2A = 2 tan A divided by (1 - tan squared A).
Three forms of cos 2A and which to use The three forms of cos 2A are not equally useful. Use 2 cos squared A - 1 when the rest of your expression is in cos. Use 1 - 2 sin squared A when the rest is in sin. The first form cos squared A - sin squared A is useful when you want to factorise (it is a difference of two squares).
R-form and harmonic identities
A common A-Level question asks you to write a sin x + b cos x in the form R sin(x + a). This is a direct use of the angle addition formula in reverse.
Expand R sin(x + a) using the formula: R sin(x + a) = R sin x cos a + R cos x sin a. Compare to a sin x + b cos x. That gives R cos a = a and R sin a = b. Squaring and adding gives R squared = a squared plus b squared, and dividing gives tan a = b over a.
The R-form is useful because it turns a sum of two trig functions into a single one, which makes solving equations like a sin x + b cos x = c much easier.
Common mistakes and how to avoid them
Examiner reports for Edexcel C3 and Pure 2 flag the same handful of mistakes on addition formula questions every year. Most of them come down to dropping or flipping a sign, or forgetting which formula matches the sign in the bracket.
Mistakes that cost easy marks Writing cos(A + B) = cos A cos B + sin A sin B (it is minus, not plus). Forgetting that the tan formula has a 1 minus or 1 plus on the denominator, not just 1. Mixing up which exact value goes with sin 30 and which with cos 30. Splitting 75 as 60 + 15 (15 is not on the unit circle) instead of 45 + 30. Writing sin 15 as a positive number times a negative quantity by accident.
Key facts to memorise for the exam
- sin(A + B) = sin A cos B + cos A sin B
- cos(A + B) = cos A cos B - sin A sin B (note the minus)
- tan(A + B) = (tan A + tan B) / (1 - tan A tan B)
- Swap the sign in the bracket and the sign in the body of the sin and tan formulae flips with it
- Swap the sign in the bracket for cos and the sign in the body flips the other way
- Set B = A to get double-angle formulae
- Three forms of cos 2A: Pick the one that matches the rest of the expression
- R sin(x + a) form comes directly from sin(A + B) in reverse
