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Question: How do you find the solution to \({{\sin }^{-1}}x={{\cos }^{-1}}x\) ?...

How do you find the solution to sin⁡−1x=cos⁡−1x{{\sin }^{-1}}x={{\cos }^{-1}}x ?

Explanation

Solution

Here we have been given an equation and we have to find the value of the unknown variable in it. Firstly we will put the given equation equal to an unknown variable. Then we will write the range of both the trigonometric functions given such that the range of the unknown variable is the intersection of the range of the two functions. Finally we will find the value for which both the functions are equal and get our desired answer.

Complete step by step solution:
We have to solve the below equation:
sin⁡−1x=cos⁡−1x{{\sin }^{-1}}x={{\cos }^{-1}}x
Let us take the above equation as follows:
sin⁡−1x=cos⁡−1x=θ{{\sin }^{-1}}x={{\cos }^{-1}}x=\theta …(1)\left( 1 \right)
Now as we know the range of trigonometric function given is as follows:
sin⁡−1x∈[−π2,π2]{{\sin }^{-1}}x\in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]
cos⁡−1x∈[0,π]{{\cos }^{-1}}x\in \left[ 0,\pi \right]
So that means,
θ∈[−π2,π2]∩[0,π]\theta \in \left[ -\dfrac{\pi }{2},\dfrac{\pi }{2} \right]\cap \left[ 0,\pi \right]
∴θ∈[0,π2]\therefore \theta \in \left[ 0,\dfrac{\pi }{2} \right]
From equation (1) we get,
sin⁡θ=cos⁡θ=x\sin \theta =\cos \theta =x….(2)\left( 2 \right)
Where θ∈[0,π2]\theta \in \left[ 0,\dfrac{\pi }{2} \right]
Next as we know sine is strictly monotonically increasing in [0,π2]\left[ 0,\dfrac{\pi }{2} \right] whereas cosine is strictly monotonically decreasing in [0,π2]\left[ 0,\dfrac{\pi }{2} \right] .
So the value of θ\theta will be the middle angle which is π4\dfrac{\pi }{4}
Therefore we have θ=π4\theta =\dfrac{\pi }{4}
So as we know that the value for sine and cosine at π4\dfrac{\pi }{4} is as follows:
sin⁡π4=cos⁡π4=22\sin \dfrac{\pi }{4}=\cos \dfrac{\pi }{4}=\dfrac{\sqrt{2}}{2}
Comparing the above value with equation (2) we get,
x=22x=\dfrac{\sqrt{2}}{2}
Hence solution of sin⁡−1x=cos⁡−1x{{\sin }^{-1}}x={{\cos }^{-1}}x is x=22x=\dfrac{\sqrt{2}}{2} .

Note:
For finding a solution for such a type of question we can directly get the value of the unknown if we know at what angle the two trigonometric values given are equal but if that is not the case then finding the range of the two values is the method we use. Here we have taken the θ=π4\theta =\dfrac{\pi }{4} as one trigonometric function is strictly increasing and other is strictly decreasing; that means the only point at which they are equal is the middle point.