Question

Find the value of the trigonometric expression  \[{\cos ^{ - 1}}\left( {\cos 12} \right) - {\sin ^{ - 1}}\left( {\sin 12} \right)\].
\[
  A.{\text{ }}0 \\
  B.{\text{ }}\pi \\
  C.{\text{ }}8\pi - 24 \\ \]
\[D.\] None of these

Answer 1

Hint:

Draw graph of \[{\cos ^{ - 1}}\left( {\cos x} \right)\] and \[{\text{si}}{{\text{n}}^{ - 1}}\left( {\sin x} \right)\].

Now as we can see that, 
We have to find the value of \[{\cos ^{ - 1}}\left( {\cos 12} \right)\] and \[{\text{si}}{{\text{n}}^{ - 1}}\left( {\sin 12} \right)\] from the above graph.
So, \[x\] will be equal to 12 in the above graphs.
Now as we can see from the above graphs of \[{\cos ^{ - 1}}\left( {\cos x} \right)\] and \[{\text{si}}{{\text{n}}^{ - 1}}\left( {\sin x} \right)\],
That principle range of \[{\cos ^{ - 1}}\left( {\cos x} \right)\] and \[{\text{si}}{{\text{n}}^{ - 1}}\left( {\sin x} \right)\] is \[\left[ {0,2\pi } \right]\].
So, we have to change 12 in terms of \[\pi \]
Now as we know that, \[\pi = 3.14\].
So, \[4\pi = 4*(3.14) = 12.56 > 12\]
\[3\pi = 3*(3.14) = 9.42 < 12\]
And, \[\dfrac{{7\pi }}{2} = \dfrac{{7*(3.14)}}{2} = 10.99 < 12\]
So, \[\dfrac{{7\pi }}{2} < 12 < 4\pi \]
So, according to the graph drawn above.
Value of \[{\cos ^{ - 1}}\left( {\cos 12} \right)\] will be \[4\pi - x\], where \[x = 12\]
And, value of \[{\text{si}}{{\text{n}}^{ - 1}}\left( {\sin 12} \right)\] will be \[x - 4\pi \], where \[x = 12\].
So, \[{\cos ^{ - 1}}\left( {\cos 12} \right) = 4\pi - 12\] (1)
And, \[{\sin ^{ - 1}}\left( {\sin 12} \right) = 12 - 4\pi \] (2)
Now, subtracting equation 1 and 2. We get,
\[{\cos ^{ - 1}}\left( {\cos 12} \right) - {\sin ^{ - 1}}\left( {\sin 12} \right) = 8\pi - 24\]
Hence, the correct option will be C.

Note:

Whenever we came up with this type of question then we first draw the graph of each trigonometric function. And then find the range in which value of \[x\] lies.
After that we can get values of trigonometric functions from the graph. Which can be then manipulated to get the required value of the given equation.