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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.

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.