Sum and difference identities · derivation ·proof

Where sin(A + B) = sin A cos B + cos A sin B comes from

Two right triangles stacked. The classical derivation drops perpendiculars from a point at angle A + B and reads the vertical leg as two pieces.

Building it step by step

1. Draw angle A + B in standard position, unit hypotenuse
   Set-up.
2. Split the vertical leg with a perpendicular at angle A
   Creates two right triangles whose angles are A and B.
3. lower piece = sin A cos B
   Adjacent leg of the B-triangle times cos B.
4. upper piece = cos A sin B
   Other leg, read with sin B.
5. sin(A + B) = sin A cos B + cos A sin B
   Add the two pieces.

How to recall it under pressure

Re-derive it from the first line rather than searching memory: start from sin(a + b) and do the one algebraic move the derivation above makes. If you need the statement itself, it is on the sine of a sum page.

sampled at 800 random anglesSource: OpenStax Precalculus, Ch. 7.2–7.3 · Paul's Online Math Notes · revised 2026-09-26 ·how we check ·accuracy policy ·report an error