Question

$-\frac{2\pi }{5}$ is the principal value of

• A. ${\cos }^{-1}\left(\cos \frac{7\pi }{5}\right)$
• B. ${\sin }^{-1}\left(\sin \frac{7\pi }{5}\right)$
• C. ${\sec }^{-1}\left(\sec \frac{7\pi }{5}\right)$
• D. none of these

Answer 1

Answer: (B)

${\sin }^{-1}\left(\sin \frac{7\pi }{5}\right)$

$\frac{-2\pi }{5}$

$=-{\sin }^{-1}\left(\sin \right(\frac{2\pi }{5}\left)\right)$

$={\sin }^{-1}(-\sin (\frac{2\pi }{5}\left)\right)$

$={\sin }^{-1}\left(\sin \right(\pi +\frac{2\pi }{5}\left)\right)$

$={\sin }^{-1}\left(\sin \right(\frac{7\pi }{5}\left)\right)$