Question

What is the value of \[\cos \left( {{\sin }^{-1}}\left( 0.5 \right) \right)\]?

Answer 1

Hint: For solving this type of questions you should know about the trigonometric values of trigonometric functions. The trigonometric values of trigonometric functions are determined at a fixed angle and that value will be permanent value at that angle. And it can be the same for one or more angles and it also can be similar for two or more functions at different – different angles.

Complete step by step solution:
According to the question we have to find the value of \[\cos \left( {{\sin }^{-1}}\left( 0.5 \right) \right)\].
As we know that the values of sin, cos or tan are different from each other but both the sin and cos are connected by each other. Because the angles of sin and cos are as \[\sin \left( 90-\theta \right)=\cos \theta \].
So, these both are dependent on each other and the values of trigonometric function are relative but opposite to each other at different – different angles.
And the \[\tan \theta \] is directly related to each other because \[\tan \theta \] is equal to the ratio of \[\sin \theta \] and \[\cos \theta \]. So, it will directly depend on both of these.
The question is \[\cos \left( {{\sin }^{-1}}\left( 0.5 \right) \right)\].
For solving this question we have to take \[{{\sin }^{-1}}\left( 0.5 \right)\] as x.
Let \[{{\sin }^{-1}}\left( 0.5 \right)=x\] then,
\[\Rightarrow \sin x=0.5\]
Since, we know that \[\cos x=\sqrt{1-{{\sin }^{2}}x}\]
So, we can write
 \[\Rightarrow \cos x=\sqrt{1-{{\left( 0.5 \right)}^{2}}}\]
\[\begin{align}
  & \Rightarrow \cos x=\sqrt{1-{{\left( \dfrac{1}{2} \right)}^{2}}} \\
 & \Rightarrow \cos x=\sqrt{1-\dfrac{1}{4}}=\dfrac{\sqrt{3}}{2} \\
 & \Rightarrow x={{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \\
\end{align}\]
Now, put this x in the question at the place of x,
\[\Rightarrow \cos \left( {{\sin }^{-1}}\left( 0.5 \right) \right)=\cos \left( {{\cos }^{-1}}\left( \dfrac{\sqrt{3}}{2} \right) \right)\]
\[\Rightarrow \cos \left( {{\sin }^{-1}}\left( 0.5 \right) \right)=\dfrac{\sqrt{3}}{2}\]
So, the value of \[\cos \left( {{\sin }^{-1}}\left( 0.5 \right) \right)\] is \[\dfrac{\sqrt{3}}{2}\].

Note: For calculating the values of trigonometric functions we always use the trigonometric value table. And if the question asks for big angles then always convert them in the form of small angles and these angles will be written the range of \[0-\pi \] and they can be multiplication of any fraction of \[\pi \]. But we can calculate the values by this easily. So, easily make them in the small form with the \[\pi \].