Question

Find the value of \[{\cos ^{ - 1}}\left( {\cos \dfrac{{7{\rm{\pi }}}}{6}} \right)\]
A) \[\dfrac{{7{\rm{\pi }}}}{6}\]
B) \[\dfrac{{{\rm{5\pi }}}}{6}\]
C) \[\dfrac{{\rm{\pi }}}{3}\]
D) \[\dfrac{{\rm{\pi }}}{6}\]

Answer 1

Hint:
Here we have to use the basic concept of trigonometry to find out the value of the given equation. So we know that whenever a function is multiplied with its own inverse then that function cancels out. So same concept we have to apply in this question also to find the value of the given equation.

Complete step by step solution:
So the given equation in the question is \[{\cos ^{ - 1}}\left( {\cos \dfrac{{7{\rm{\pi }}}}{6}} \right)\]
We can clearly see that in the given equation cos function is multiplying with its own cos inverse function. A function cancels out when it is multiplied with its own inverse function. Similarly in this equation also the cos function gets canceled out by its cos inverse function then we will get the remaining terms as the value of the given equation.
\[{\cos ^{ - 1}}\left( {\cos \dfrac{{7{\rm{\pi }}}}{6}} \right) = \dfrac{{7{\rm{\pi }}}}{6}\]
We can clearly see that \[\dfrac{{7{\rm{\pi }}}}{6}\] is the only remaining term from the equation when the cos function gets cancelled out its cos inverse function.
 Therefore, we get \[\dfrac{{7{\rm{\pi }}}}{6}\] as the value of the given equation.
Hence, \[{\cos ^{ - 1}}\left( {\cos \dfrac{{7{\rm{\pi }}}}{6}} \right)\]is equal to \[\dfrac{{7{\rm{\pi }}}}{6}\]

So, option A is correct.

Note:
Inverse of a function is the function which is totally opposite of the main function. The relation between the sides and angles of a right triangle is the basis for trigonometry. Right Triangle is a triangle where one of its interior angles is a right angle (90 degrees). The side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called base.
We have to remember all the trigonometry formulas
\[{\rm{sin A = }}\dfrac{{{\text{side opposite to angle A}}}}{{{\text{hypotenuse}}}}\], \[{\rm{cos A = }}\dfrac{{{\text{side adjacent to angle A}}}}{{{\text{hypotenuse}}}}\], \[{\rm{tan A = }}\dfrac{{{\text{side opposite to angle A}}}}{{{\text{side adjacent to angle A}}}}\], \[{\rm{cot A = }}\dfrac{1}{{{\rm{tan A}}}}\], \[{\rm{sec A = }}\dfrac{1}{{{\rm{cos A}}}}\], \[{\rm{cosec A = }}\dfrac{1}{{{\rm{sin A}}}}\]