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# Find the value of ${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)$.  Answer Verified
Hint: Simplify $\cos 1540{}^\circ$ so that you can use the property ${{\cos}^{-1}}\left( \cos x \right)=x$. For this, use the property that a trigonometric operation of the form $\cos \left( 360{}^\circ \times n+x \right)$ can be written as $\cos \left( x \right)$. Next use the property ${{\cos }^{-1}}\left( \cos x \right)=x$ on the simplified expression to arrive at the final answer.

Complete step by step solution:
In this question, we need to find the value of ${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)$.
We first need to identify that the range of the function ${{\cos }^{-1}}\left( x \right)$ is between $0{}^\circ$ and $180{}^\circ$.
In our question, we are given $1540{}^\circ$ which is not in this range. So, we cannot
directly write ${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)=1540{}^\circ$. So, we need to
simplify $1540{}^\circ$.
To find this value, we will first evaluate $\cos 1540{}^\circ$ and then we will come to the inverse part.
First, let us simplify $\cos 1540{}^\circ$
We can write 1540 as:
$1540=360\times 4+100$
So, we can write $\cos 1540{}^\circ$ as the following:
$\cos 1540{}^\circ =\cos \left( 360{}^\circ \times 4+100{}^\circ \right)$
Now, we know the property that a trigonometric operation of the form $\cos \left( 360{}^\circ \times n+x \right)$ can be written as $\cos \left( x \right)$.
Here, in this question we have n = 4 and x = 100.
Using this property, we can write the above expression as:
$\cos 1540{}^\circ =\cos \left( 360{}^\circ \times 4+100{}^\circ \right)$
$\cos 1540{}^\circ =\cos 100{}^\circ$
Now, we will come to the inverse part.
We know the property that for an angle x, if the measure of angle x is greater than or equal
to $0{}^\circ$ and less than or equal to $180{}^\circ$ , then the expression ${{\cos }^{- 1}}\left( \cos x \right)$ can be written as x
i.e. ${{\cos }^{-1}}\left( \cos x \right)=x$ for $0{}^\circ \le x\le 180{}^\circ$
Now, since in our question $100{}^\circ$ satisfies the condition of being greater than or
equal to $0{}^\circ$ and less than or equal to $180{}^\circ$ , we can use the above
property on it.
We will use this property to calculate ${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)$
${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)={{\cos }^{-1}}\left( \cos 100{}^\circ \right)$

${{\cos }^{-1}}\left( \cos 100{}^\circ \right)=100{}^\circ$
Hence, ${{\cos }^{-1}}\left( \cos 1540{}^\circ \right)=100{}^\circ$
This is our final answer.

Note: In this question, it is very important to identify that the range of the function ${{\cos}^{-1}}\left( x \right)$ is between $0{}^\circ$ and $180{}^\circ$. In our question, we are given $1540{}^\circ$ which is not in this range. So, we cannot directly write ${{\cos }^{-1}}\left(\cos 1540{}^\circ \right)=1540{}^\circ$. This would be wrong. So, we need to simplify $1540{}^\circ$ to a smaller number such that it can be within the range.
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