Question

Solve for ‘x’: $\cos \left( {{\sin }^{-1}}x \right)=\dfrac{1}{7}$.

Answer 1

In our problem, the inverse of sine function is being operated by cosine function. These two not being inverse of each other cannot cancel out. So, our main aim is to convert one of them into the other function. This can be achieved by the method of substitution. We shall proceed in this manner to get our answer.

Complete step by step solution:
We have to solve the expression: $\cos \left( {{\sin }^{-1}}x \right)=\dfrac{1}{7}$. This can be solved by converting either cosine into sine function or sine into cosine. It can be done by the method of substitution. This is done as follows:
Let us substitute: ${{\sin }^{-1}}x\to p$. Then, on solving this substitution, we get:
$\begin{aligned} & \Rightarrow {{\sin }^{-1}}x=p \\\ & \Rightarrow x=\sin p \\\ & \Rightarrow {{x}^{2}}={{\sin }^{2}}p \\\ & \Rightarrow 1-{{x}^{2}}=1-{{\sin }^{2}}p \\\ & \Rightarrow 1-{{x}^{2}}={{\cos }^{2}}p \\\ \end{aligned}$
Let us say this is equation number (1). So, we have:
$\Rightarrow 1-{{x}^{2}}={{\cos }^{2}}p$ ......... (1)

Now, putting the value of ${{\sin }^{-1}}x$ as ‘p’, in our original equation, we get the new equation as:
$\Rightarrow \cos \left( p \right)=\dfrac{1}{7}$
Squaring terms on both sides of the equation, we get:
$\Rightarrow {{\cos }^{2}}p={{\left( \dfrac{1}{7} \right)}^{2}}$
Let us say this is equation number (2). So, we have:
$\Rightarrow {{\cos }^{2}}p={{\left( \dfrac{1}{7} \right)}^{2}}$ ......... (2)

Now, putting the value of ${{\cos }^{2}}p$ from equation number (1) into equation number (2), we get the new expression only in variable ‘x’ as:
$\begin{aligned} & \Rightarrow 1-{{x}^{2}}={{\left( \dfrac{1}{7} \right)}^{2}} \\\ & \Rightarrow 1-{{x}^{2}}=\left( \dfrac{1}{49} \right) \\\ & \Rightarrow {{x}^{2}}=1-\dfrac{1}{49} \\\ & \Rightarrow {{x}^{2}}=\dfrac{48}{49} \\\ \end{aligned}$
Taking square roots on both sides of the equation, we get:
$\begin{aligned} & \Rightarrow x=\sqrt{\dfrac{48}{49}} \\\ & \therefore x=\dfrac{4\sqrt{3}}{7} \\\ \end{aligned}$
Hence, the value of ‘x’ comes out to be $\dfrac{4\sqrt{3}}{7}$.

Note: Whenever there are two different types of trigonometric identities involved in a problem, the best approach to solve them is by converting one of the quantities into another one. This results in the expression containing only a type of trigonometric entity and it becomes easier to solve the equation. Also, one should always cross-verify the answer obtained to ensure correction.