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QJEEM24Ma255b: JEE Main 2024 Mathematics - Class 12 JEEM24Ma255b Solution: Let the inverse trigonometric functions take principal values.…

Let the inverse trigonometric functions take principal values. The number of real solutions of the equation 2sin⁡−1x+3cos⁡−1x=2π52 \sin ^{-1} x+3 \cos ^{-1} x=\frac{2 \pi}{5} , is __________.

Step-by-Step Verified Solution

amp;2sin⁡−1x+3cos⁡−1x=2π5amp;π2+cos⁡−1x=2π5amp;cos⁡−1x=2π5−π2amp;cos⁡−1x=−π10\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}

Which is not possible as cos⁡−1x∈[0,π]\cos ^{-1} x \in[0, \pi]

∴\therefore \quad No solution

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