Question

The principal value of \[{\cos ^{ - 1}}\left[ {\dfrac{1}{{\sqrt 2 }}\left( {\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right)} \right]\] is:
A). \[\dfrac{{3\pi }}{{20}}\]
B). \[\dfrac{{7\pi }}{{20}}\]
C). \[\dfrac{{7\pi }}{{10}}\]
D). None of these

Answer 1

Hint: In the given question, we have been given a trigonometric function. This function consists of the inverse of the basic trigonometric function with the argument as a combination of more basic trigonometric functions. We have to find the principal value of this trigonometric function. We are going to solve it by making the given constant as the value equal to trigonometric functions. Then we are going to apply the appropriate formula to solve for the given question.

Formula used:
We are going to use the formula of sum of cosine function, which is,
\[\cos \left( {A + B} \right) = \cos A\cos B - \sin A\sin B\]

Complete step by step solution:
The given expression is \[{\cos ^{ - 1}}\left[ {\dfrac{1}{{\sqrt 2 }}\left( {\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right)} \right] = {\cos ^{ - 1}}\left[ {\dfrac{1}{{\sqrt 2 }}\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \dfrac{1}{{\sqrt 2 }}\sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right]\].
Now, \[\cos \dfrac{\pi }{4} = \sin \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}\].
So,
\[{\cos ^{ - 1}}\left[ {\dfrac{1}{{\sqrt 2 }}\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \dfrac{1}{{\sqrt 2 }}\sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right] = {\cos ^{ - 1}}\left[ {\cos \dfrac{\pi }{4}\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \sin \dfrac{\pi }{4}\sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right]\]
Now, we know the formula of sum of cosine function, which is,
\[\cos \left( {A - B} \right) = \cos A\cos B - \sin A\sin B\]
So, we have,
\[{\cos ^{ - 1}}\left[ {\cos \dfrac{\pi }{4}\cos \left( {\dfrac{{9\pi }}{{10}}} \right) - \sin \dfrac{\pi }{4}\sin \left( {\dfrac{{9\pi }}{{10}}} \right)} \right] = {\cos ^{ - 1}}\left[ {\cos \left( {\dfrac{\pi }{4} + \dfrac{{9\pi }}{{10}}} \right)} \right] = {\cos ^{ - 1}}\left[ {\cos \left( {\dfrac{{23\pi }}{{20}}} \right)} \right]\]
Now, \[{\cos ^{ - 1}}\left[ {\cos \left( {\dfrac{{23\pi }}{{20}}} \right)} \right] = {\cos ^{ - 1}}\left[ {\cos \left( {\pi + \dfrac{{3\pi }}{{20}}} \right)} \right] = {\cos ^{ - 1}}\left[ {\cos \left( {\dfrac{{3\pi }}{{20}}} \right)} \right]\]
Hence, the value of the given expression is,
\[\dfrac{{3\pi }}{{20}}\]
Thus, the correct option is A.

Note: In this question, we were given a trigonometric function. This function consisted of the inverse of the basic trigonometric function with the argument as a combination of more basic trigonometric functions. We had to find the principal value of this trigonometric function. We solved it by substituting the value of the constant equal to the value of the corresponding trigonometric function so that it becomes equal to the trigonometric function whose inverse is being taken. And then we just put the value into the formula and got our answer.