Question

How do you find the value of $\cos \left( {{{\cos }^{ - 1}}\left( {\dfrac{3}{4}} \right)} \right)$?

Answer 1

To find the value of such a question, we will first assume an angle that will be the value of ${\cos ^{ - 1}}\left( {\dfrac{3}{4}} \right)$. On assuming the angle, we will then place the angle within the cosine function. Since the cosine function gives a value for an angle, we will get the desired value of the cosine function.

Complete step by step answer:
Given, $\cos \left( {{{\cos }^{ - 1}}\left( {\dfrac{3}{4}} \right)} \right) - - - \left( 1 \right)$.
We know, every inverse trigonometric function gives an angle.
So, ${\cos ^{ - 1}}\left( {\dfrac{3}{4}} \right)$ will also give us an angle.
So, let us assume that the angle is $\theta$.
So, we can write it as,
$\theta = {\cos ^{ - 1}}\left( {\dfrac{3}{4}} \right) - - - \left( 2 \right)$.
Now, substituting this value in the given function $\left( 1 \right)$, we get, $\cos \left( \theta \right)$.
Now, we can write $\theta$ from $\left( 2 \right)$ as an angle that gives $\dfrac{3}{4}$ as a value on taking the cosine function of the angle.
So, on substituting the angle on to the main function, we get,
$\cos \left( \theta \right)$
This is also the cosine function of the angle $\theta$.
So, as we have already seen above, the cosine function of the angle $\theta$, will give $\dfrac{3}{4}$ as the value.
So, $\cos \left( \theta \right)$ will give $\dfrac{3}{4}$ as the value.
That is, $\cos \left( \theta \right) = \dfrac{3}{4}$.
Substituting the value of $\theta$ from $\left( 2 \right)$ in the above equation, we get,
$\cos \left( {{{\cos }^{ - 1}}\left( {\dfrac{3}{4}} \right)} \right) = \dfrac{3}{4}$.
Therefore, the value of $\cos \left( {{{\cos }^{ - 1}}\left( {\dfrac{3}{4}} \right)} \right)$ is $\dfrac{3}{4}$.

Note: The inverse trigonometric function gives angles as the output for every function. Every inverse trigonometric function has its own principal range and its own domain. But since the trigonometric functions are repetitive, they can also be represented as general forms over a much vast range. The principal range of inverse cosine function is $\left[ {0,\pi } \right]$.