From the sine addition formula, prove sin A+sin B=2sin((A+B)/2)cos((A−B)/2).

01 / Standard#Your turn

Official paper · jm02-2021 · 4(a) · PDF 6

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Write A=u+v and B=u−v.

Hint 2

Add the expansions so the cos u sin v terms cancel.

Worked solution

Define the half-sum and half-difference.
$u=(A+B)/2,\quad v=(A-B)/2$
Expand and add.
$\sin(u+v)+\sin(u-v)=2\sin u\cos v$
Substitute the definitions back.
$\sin A+\sin B=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$

The sum-to-product identity follows.

Checks and common pitfalls: The two expansions have opposite signs on cos u sin v.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Define the half-sum and half-difference.
$u=(A+B)/2,\quad v=(A-B)/2$
Expand and add.
$\sin(u+v)+\sin(u-v)=2\sin u\cos v$
Substitute the definitions back.
$\sin A+\sin B=2\sin\frac{A+B}{2}\cos\frac{A-B}{2}$

Think first. Reveal a hint when the class is ready.

2021 JM02