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If $\sin 3 A=1$, then how many distinct values can $\sin A$ assume?
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The correct answer is:
2
Let $\sin 3 A=1$
$\Rightarrow 3 \sin A-4 \sin ^{3} A=1$
$\Rightarrow 4 \sin ^{3} A-3 \sin A+1=0$
$\Rightarrow(\sin A+1)\left(4 \sin ^{2} A-4 \sin A+1\right)=0$
$\Rightarrow(\sin A+1)(2 \sin A-1)^{2}=0$
$\Rightarrow \sin A=-1$ or $\frac{1}{2}$
Hence, $\sin A$ can take two distinct values
$\Rightarrow 3 \sin A-4 \sin ^{3} A=1$
$\Rightarrow 4 \sin ^{3} A-3 \sin A+1=0$
$\Rightarrow(\sin A+1)\left(4 \sin ^{2} A-4 \sin A+1\right)=0$
$\Rightarrow(\sin A+1)(2 \sin A-1)^{2}=0$
$\Rightarrow \sin A=-1$ or $\frac{1}{2}$
Hence, $\sin A$ can take two distinct values
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