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Question

Mathematics Question on Trigonometric Equations

If sin⁡−1 x+cos⁡−1 y=2π5\sin^{-1} \,x + \cos^{-1} \,y = \frac{2 \pi}{5} , then cos⁡−1 x+sin⁡−1 y\cos^{-1} \,x + \sin^{-1}\, y is

A

2π5\frac{2 \pi}{5}

B

3π5\frac{3 \pi}{5}

C

4π5\frac{4 \pi}{5}

D

3π10\frac{3 \pi}{10}

Answer

3π5\frac{3 \pi}{5}

Explanation

Solution

Given sin⁡−1 x+cos⁡−1 y=2π5\sin^{-1} \,x + \cos^{-1}\,y = \frac{2\pi}{5}
⇒(π2−cos⁡−1x)+(π2−sin⁡−1y)=2π5\Rightarrow \left(\frac{\pi}{2} -\cos^{-1}x\right)+\left(\frac{\pi}{2} -\sin^{-1}y \right) = \frac{2\pi}{5}
⇒cos⁡−1x+sin⁡−1y=π−2π5=3π5\Rightarrow \cos^{-1}x + \sin^{-1}y =\pi - \frac{2\pi}{5} = \frac{3\pi}{5}