Given A+B+C=π, prove sin A+sin B+sin C=4cos(A/2)cos(B/2)cos(C/2).

01 / Standard#Your turn

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

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use part (a) on sin A+sin B.

Hint 2

Replace (A+B)/2 by (π−C)/2 and factor cos(C/2).

Worked solution

Combine the first two terms and the double-angle form of the third.
$\sin A+\sin B+\sin C=2\cos(C/2)\left[\cos\frac{A-B}{2}+\sin(C/2)\right]$
The angle-sum condition converts sine into cosine.
$\sin(C/2)=\cos\frac{A+B}{2}$
Expand the two cosines; the sine-product terms cancel.
$\cos\frac{A-B}{2}+\cos\frac{A+B}{2}=2\cos(A/2)\cos(B/2)$
Multiply the remaining factors, without dividing by a potentially zero cosine.
$\sin A+\sin B+\sin C=4\cos(A/2)\cos(B/2)\cos(C/2)$

The identity is proved whenever A+B+C=π.

Checks and common pitfalls: The proof uses no division by cos(C/2), so zero-cosine cases remain included.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Combine the first two terms and the double-angle form of the third.
$\sin A+\sin B+\sin C=2\cos(C/2)\left[\cos\frac{A-B}{2}+\sin(C/2)\right]$
The angle-sum condition converts sine into cosine.
$\sin(C/2)=\cos\frac{A+B}{2}$
Expand the two cosines; the sine-product terms cancel.
$\cos\frac{A-B}{2}+\cos\frac{A+B}{2}=2\cos(A/2)\cos(B/2)$
Multiply the remaining factors, without dividing by a potentially zero cosine.
$\sin A+\sin B+\sin C=4\cos(A/2)\cos(B/2)\cos(C/2)$

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

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