Assertion (A) : Principal value of ^ -1 ( (2 3 ) ) is 3 . Reason … - Vidyadip

MCQ

Assertion (A) : Principal value of $\sin ^{-1}\left(\sin \left(\frac{2 \pi}{3}\right)\right)$ is $\frac{\pi}{3}$.
Reason (R) : Principal value branch of $\sin ^{-1}$ function is $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$.

✓
Both (A) and (R) are true and (R) is the correct explanation of (A).
B
Both (A) and (R) are true but (R) is not the correct explanation of (A).
C
(A) is true but (R) is false.
D
(A) is false but (R) is true.

✓

Answer

Correct option: A.

Both (A) and (R) are true and (R) is the correct explanation of (A).

(a) : Let $y=\sin ^{-1}\left(\frac{2 \pi}{3}\right) \sin ^{-1}(\sin (\pi-\pi / 3))$
$=\sin ^{-1}\left(\sin \frac{\pi}{3}\right)=\sin ^{-1}\left(\frac{\sqrt{3}}{2}\right)$
$=\frac{\pi}{3} \quad\left[\because \sin ^{-1}:[-1,1] \rightarrow\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\right]$

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