Question

If $\theta_1=\sin ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{1}{3}\right)$ and $\theta_2=\cos ^{-1}\left(\frac{4}{5}\right)+\cos ^{-1}\left(\frac{1}{3}\right)$, then

• A. $\theta_1>\theta_2$
• B. $\theta_1=\theta_2$
• C. $\theta_1<\theta_2$
• D. none of these

Answer 1

Answer: (C)

$\theta_1=\tan ^{-1} \frac{4}{3}+\tan ^{-1} \frac{1}{2 \sqrt{2}}=\tan ^{-1} \frac{8 \sqrt{2}+3}{6 \sqrt{2}-4}<\frac{\pi}{2}$
$\theta_2=\frac{\pi}{2}-\sin ^{-1} \frac{4}{5}+\frac{\pi}{2}-\sin ^{-1} \frac{1}{3}=\pi-\theta_1>\frac{\pi}{2}$