Question

Let f (x) = `e^(cos^-1){sin(x+pi/3}.`
Then, f (8π/9) = 

Options

• e^5π/18

•  e^13π/18

• e^−2π/18

• none of these

Answer 1

(b) e^13π/18

Given: \[f\left( x \right) = e^{\cos^{- 1}} \left\{ \sin\left( x + \frac{\pi}{3} \right) \right\}\]
Then,

\[f\left( \frac{8\pi}{9} \right) = e^{\cos^{- 1}} \left\{ \sin\left( \frac{8\pi}{9} + \frac{\pi}{3} \right) \right\} \]
\[ = e^{\cos^{- 1}} \left\{ \sin\left( \frac{11\pi}{9} \right) \right\} \]
\[ = e^{\cos^{- 1}} \left\{ \cos\left( \frac{\pi}{2} + \frac{13\pi}{18} \right) \right\} \left[ \because \cos\left( \frac{\pi}{2} + \theta \right) = \sin\theta \right]\]
\[ = e^{\cos^{- 1}} \left\{ \cos\left( \frac{13\pi}{18} \right) \right\} \]
\[ = e^\frac{13\pi}{18}\]