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Evaluate the following: ${\cos ^{ - 1}}\left( {\cos 12} \right)$.

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Last updated date: 26th Feb 2024
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Hint: Here we will use the value of inverse trigonometric cosine function of the form ${\cos ^{ - 1}}\left( {\cos x} \right) = x$ when $x$ lies in the interval $\left[ {0,\pi } \right]$. So, we will check for the quadrant in which the argument of the cosine function lies to get the exact value of the function.

Complete step by step solution:
In the question, we have to find the value of the expression ${\cos ^{ - 1}}\left( {\cos 12} \right)$.
Here we will use the value of inverse trigonometric cosine function of the form ${\cos ^{ - 1}}\left( {\cos x} \right) = x$ when $x$ lies in the interval $\left[ {0,\pi } \right]$.
But when $x$ lies in the interval $\dfrac{{3\pi }}{2} < x < 2\pi $, then ${\cos ^{ - 1}}\left( {\cos x} \right) = 2\pi - x$. So, this is an important concept that is to be used here.
Now, in the problem we have $\cos 12$ which has the argument as 12 which lies in the interval,
$ \Rightarrow \dfrac{{7\pi }}{2} < 12 < 4\pi $
So, to bring that in the required interval of $\left[ {0,\pi } \right]$ we can write 12 as,
$ \Rightarrow 4\pi - 12$
Also, we know that,
$ \Rightarrow \cos \left( {4\pi - 12} \right) = \cos 12$
Now, we have the expression ${\cos ^{ - 1}}\left( {\cos \left( {4\pi - 12} \right)} \right)$ for the given expression ${\cos ^{ - 1}}\left( {\cos 12} \right)$.
Here, $\left( {4\pi - 12} \right)$ lies in the interval $\left[ {0,\pi } \right]$ and that can be shown below:
$ \Rightarrow \dfrac{{7\pi }}{2} < 12 < 4\pi $
Multiply by -1 on all sides,
$ \Rightarrow - \dfrac{{7\pi }}{2} > - 12 > - 4\pi $
Add $4\pi $ on all sides,
$ \Rightarrow 4\pi - \dfrac{{7\pi }}{2} > 4\pi - 12 > 4\pi - 4\pi $
Subtract the terms to get the interval,
$ \Rightarrow \dfrac{\pi }{2} > 4\pi - 12 > 0$
Now, replace 12 with $\left( {4\pi - 12} \right)$ in the expression,
$ \Rightarrow {\cos ^{ - 1}}\left( {\cos 12} \right) = {\cos ^{ - 1}}\left( {\cos \left( {4\pi - 12} \right)} \right)$
Use the formula, ${\cos ^{ - 1}}\left( {\cos x} \right) = x$ to get the value,
$\therefore {\cos ^{ - 1}}\left( {\cos \left( {4\pi - 12} \right)} \right) = 4\pi - 12$

Hence, the value of the expression ${\cos ^{ - 1}}\left( {\cos 12} \right)$ is $4\pi - 12$.

Note: We have to be careful in finding the value of the inverse trigonometric function. It is important to check the quadrant in which the argument of the trigonometric function lies. So \[{\cos ^{ - 1}}\left( {\cos x} \right) = x\] is not true for all $x$, but this is only true if the argument $x$ lies in the interval $\left[ {0,\pi } \right]$.