Trigonometry · step 6 of 13

Angle-sum, difference & double-angle formulas

sin(A + B) = sin A cos B + cos A sin B, and cos(A + B) = cos A cos B − sin A sin B — note that the cosine formula flips the sign, which is the detail most often forgotten. Setting B = A gives the double-angle formulas, and cos 2A has three equivalent forms, each useful in a different context.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Learn the sign conventionSine keeps the sign of the operation; cosine reverses it. cos(A+B) has a MINUS.
Use sums to get exact values75° = 45° + 30°, and 15° = 45° − 30°, so exact values extend beyond the standard angles.
Pick the right form of cos 2Acos²A − sin²A, or 1 − 2sin²A, or 2cos²A − 1. Choose whichever matches the other terms present.
Reverse them for power reductionsin²A = (1 − cos 2A)/2 and cos²A = (1 + cos 2A)/2 — essential for integration.

Worked example

Find the exact value of cos 75°.

Write 75° = 45° + 30°.
cos(A+B) = cos A cos B − sin A sin B.
= (√2/2)(√3/2) − (√2/2)(1/2).
= √6/4 − √2/4.

Answer. cos 75° = (√6 − √2)/4 ≈ 0.2588.

Where marks get dropped

These are the specific errors that cost credit on angle-sum, difference & double-angle formulas questions — QED's rubric penalises each of them separately.

Using a plus sign in cos(A+B). The cosine formula reverses the sign — this single error accounts for most lost marks here.
Assuming sin(A+B) = sin A + sin B. It is not, and testing A = B = 30° shows it immediately.
Choosing an inconvenient form of cos 2A and creating extra work. If the expression already has sin²A, use 1 − 2sin²A.

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Angle-sum, difference & double-angle formulas — frequently asked questions

How do I remember the sign in cos(A+B)?

Test it: cos(90°) = 0 with A = B = 45° gives (√2/2)² − (√2/2)² = 0 ✓, whereas a plus sign would give 1. One check settles it forever.

What is the tangent addition formula?

tan(A+B) = (tan A + tan B)/(1 − tan A tan B). It follows by dividing the sine formula by the cosine one.

Why are the power-reduction forms important?

Because ∫sin²x dx cannot be done directly, but ∫(1 − cos 2x)/2 dx is immediate. Every integral of an even power of sine or cosine uses them.

The rest of Trigonometry

Triangles, the unit circle, identities and periodic functions. Each subtopic below has its own method, worked example and mark-losing traps.

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