Question

find x if 4sin inverse x= π- cos inverse x​

Answer 1

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Answer 2

Given :

$\rm{4{\sin}^{-1}x=\pi-{\cos}^{-1}x}$

To Find :

The value of x

Properties of inverse trigonometric functions:

$1)\rm{{\sin}^{-1}x+{\cos}^{-1}x=\dfrac{\pi}{2}}$

$2)\rm{{\tan}^{-1}x+{\cot}^{-1}x=\dfrac{\pi}{2}}$

$3)\rm{{\sec}^{-1}x+{\cosec}^{-1}x=\dfrac{\pi}{2}}$

Solution :

We have to find the value of x

$\rm{4{\sin}^{-1}x=\pi-{\cos}^{-1}x}$

We know that

$\rm{{\sin}^{-1}x+{\cos}^{-1}x=\dfrac{\pi}{2}}$

$\implies\rm{4{\sin}^{-1}x=\pi-(\dfrac{\pi}{2}-{\sin}^{-1}x)}$

$\implies\rm{4{\sin}^{-1}x=\pi-\dfrac{\pi}{2}+{\sin}^{-1}x)}$

$\implies\rm{4{\sin}^{-1}x-{\sin}^{-1}x=\pi-\dfrac{\pi}{2}}$

$\implies\rm{3{\sin}^{-1}x=\dfrac{\pi}{2}}$

$\implies\rm{{\sin}^{-1}x=\dfrac{\pi}{6}}$

$\implies\rm{\sin\dfrac{\pi}{6}=x}$

$\implies\rm{x=\dfrac{1}{2}}$

Therefore, The value of x is 1/2 .