For triangle ABC with the stated conditions, find sin²C.

01 / Standard#Your turn

A+B+C=\pi,\quad\sin(C-A)=1,\quad\cos B=2\sqrt2/3

Official paper · jm01-2021 · II.5(a) · PDF 4

Skills and prerequisite lessons

Working and explanation

BUILD THE REASONING

Hint 1

The triangle-angle ranges make C−A=π/2.

Hint 2

Use the angle sum to express 2C in terms of B.

Worked solution

The cosine determines positive sine of the interior angle B.
$\sin B=\sqrt{1-8/9}=1/3$
Combine C−A=π/2 with A+B+C=π.
$2C=3\pi/2-B\Rightarrow\cos2C=-\sin B=-1/3$
Use the power-reduction identity.
$\sin^2C=(1-\cos2C)/2=(1+1/3)/2=2/3$

sin²C=2/3.

Checks and common pitfalls: The requested expression is sine squared, not sine of twice the angle.

Reasoning checklist · self / teacher assessment

Teaching assessment checklist, independently authored. Use the original paper for official marks.
The cosine determines positive sine of the interior angle B.
$\sin B=\sqrt{1-8/9}=1/3$
Combine C−A=π/2 with A+B+C=π.
$2C=3\pi/2-B\Rightarrow\cos2C=-\sin B=-1/3$
Use the power-reduction identity.
$\sin^2C=(1-\cos2C)/2=(1+1/3)/2=2/3$

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

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