In a nondegenerate triangle, determine cos C from the half-angle relation.

01 / Standard#Your turn

\sin(A+B)=6\sin^2\frac C2

Official paper · jm01-2025 · II.3(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

Use A+B=π−C.

Hint 2

Divide by the positive sin(C/2).

Worked solution

Convert sin C to half-angle factors.
$2\sin\frac C2\cos\frac C2=6\sin^2\frac C2\implies\tan\frac C2=\frac13$
Apply the tangent half-angle expression for cosine.
$\cos C=\frac{1-\tan^2(C/2)}{1+\tan^2(C/2)}=\frac45$

cos C=4/5.

Checks and common pitfalls: C=0 is excluded because C is an interior angle of a nondegenerate triangle.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
Convert sin C to half-angle factors.
$2\sin\frac C2\cos\frac C2=6\sin^2\frac C2\implies\tan\frac C2=\frac13$
Apply the tangent half-angle expression for cosine.
$\cos C=\frac{1-\tan^2(C/2)}{1+\tan^2(C/2)}=\frac45$

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

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