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Question
Let f (x) = `e^(cos^-1){sin(x+pi/3}.`
Then, f (8π/9) =
Options
e5π/18
e13π/18
e−2π/18
none of these
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Solution
(b) e13π/18
Given: \[f\left( x \right) = e^{\cos^{- 1}} \left\{ \sin\left( x + \frac{\pi}{3} \right) \right\}\]
Then,
\[f\left( \frac{8\pi}{9} \right) = e^{\cos^{- 1}} \left\{ \sin\left( \frac{8\pi}{9} + \frac{\pi}{3} \right) \right\} \]
\[ = e^{\cos^{- 1}} \left\{ \sin\left( \frac{11\pi}{9} \right) \right\} \]
\[ = e^{\cos^{- 1}} \left\{ \cos\left( \frac{\pi}{2} + \frac{13\pi}{18} \right) \right\} \left[ \because \cos\left( \frac{\pi}{2} + \theta \right) = \sin\theta \right]\]
\[ = e^{\cos^{- 1}} \left\{ \cos\left( \frac{13\pi}{18} \right) \right\} \]
\[ = e^\frac{13\pi}{18}\]
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