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 (integer)
0
Solution
<p>$$\begin{aligned}
& 2 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5} \\
& \frac{\pi}{2}+\cos ^{-1} x=\frac{2 \pi}{5} \\
& \cos ^{-1} x=\frac{2 \pi}{5}-\frac{\pi}{2} \\
& \cos ^{-1} x=\frac{-\pi}{10}
\end{aligned}$$</p>
<p>Which is not possible as $\cos ^{-1} x \in[0, \pi]$</p>
<p>$\therefore \quad$ No solution</p>
About this question
Subject: Mathematics · Chapter: Inverse Trigonometric Functions · Topic: Properties and Identities
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