Let the inverse trigonometric functions take principal values.

Question

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation $$2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5}$$, is \_\_\_\_\_\_\_\_\_\_.

Answer: 0

Accepted Answer

$\begin{gathered} 2{\sin }^{-1}x+3{\cos }^{-1}x=\frac{2\pi }{5} \ \frac{\pi }{2}+{\cos }^{-1}x=\frac{2\pi }{5} \ {\cos }^{-1}x=\frac{2\pi }{5}-\frac{\pi }{2} \ {\cos }^{-1}x=\frac{-\pi }{10} \end{gathered}$

Which is not possible as ${\cos }^{-1}x\in [0,\pi ]$

$	herefore \,$ No solution

Solutions · 1