Question

If \( \sum_{i=1}^{300} \cos ^{-1} x_{i}=0 \). Then find the value of \( \sum_{i=1}^{300} \sin ^{-1} x_{i} \) 

(1) 0 

(2) \( 150 \pi \) 

(3) 150 

(4) \( 151 \pi \)

Answer 1

Correct option is (2) 150π

\(\sum^{300}_{i =1} cos^{-1}x_i = 0\)    (Given)

\(\therefore\sum^{300}_{i =1} \left(\frac{\pi}{2} - sin^{-1}x_i\right)= 0\)           \(\left(\because cos^{-1}x = \frac{\pi}{2} - sin^{-1}x\right)\)

⇒ \(\frac{\pi}{2}\times 300 - \sum^{300}_{i =1}sin^{-1}x_i = 0\)    \(\left(\because \sum^{300}_{i = 1} = 300\right)\) 

⇒ \(\sum^{300}_{i = 1 }sin^{-1}x_i = 150 \pi\)