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)

${\theta }_1={\tan }^{-1}\frac{4}{3}+{\tan }^{-1}\frac{1}{2\sqrt{2}}={\tan }^{-1}\frac{8\sqrt{2}+3}{6\sqrt{2}-4}<\frac{\pi }{2}$
${\theta }_2=\frac{\pi }{2}-{\sin }^{-1}\frac{4}{5}+\frac{\pi }{2}-{\sin }^{-1}\frac{1}{3}=\pi -{\theta }_1>\frac{\pi }{2}$