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Question

Mathematics Question on Trigonometric Equations

The principal value of sin−1(sin5π3)sin^{-1} \left(sin \frac{5\pi}{3}\right) is

A

−5π3-\frac{5\pi}{3}

B

5π3\frac{5\pi}{3}

C

−π3-\frac{\pi}{3}

D

4π3\frac{4\pi}{3}

Answer

−π3-\frac{\pi}{3}

Explanation

Solution

Let θ=sin−1[sin5π3]\theta = sin^{-1}\left[sin\frac{5\pi}{3}\right]
⇒sin θ=sin5π3=sin[2π−π3]\Rightarrow sin\,\theta = sin\frac{5\pi }{3} = sin\left[2\pi-\frac{\pi}{3}\right]
⇒sin θ=−sinπ3=sin(−π3)\Rightarrow sin\,\theta = -sin\frac{\pi }{3} = sin \left(\frac{-\pi }{3}\right)
(∵ sin (−θ)= − sin θ)\left(\because \,sin\, \left(- \theta \right)= \,- \,sin \,\theta\right)
Therefore, principal value of sin−1sin^{-1}
[sin5π3]\left[sin\frac{5\pi }{3}\right] is −π3\frac{-\pi }{3}, as principal value of sin−sin^{-}
1 x^{1} \,x lies between −π2\frac{-\pi }{2} and π2.\frac{\pi }{2}.