Solveeit Logo
Login

Question

Question: How do you simplify \[{\cos ^{ - 1}}\left( {\dfrac{{\sqrt 2 }}{2}} \right)\]?...

How do you simplify cos⁡−1(22){\cos ^{ - 1}}\left( {\dfrac{{\sqrt 2 }}{2}} \right)?

Explanation

Solution

In the given question, we have been given a trigonometric function. This trigonometric function is raised to some negative power. Also, this trigonometric function has a constant as its argument. We just need to know what this negative power means. Any trigonometric function raised to this negative power means that it is the inverse of that trigonometric function. So, we have to find the inverse of the given trigonometric function. By finding the inverse it means that we have to find the angle which when put into the given trigonometric function, yields the same value as is given in the inverse of the trigonometric function.

Complete step by step answer:
The given trigonometric function is cos⁡−1(22){\cos ^{ - 1}}\left( {\dfrac{{\sqrt 2 }}{2}} \right). When a trigonometric function is raised to a negative power, it means that we have to find the inverse of the given trigonometric function. By finding the inverse it means that we have to calculate the angle which gives the value which is inside the given inverse of the trigonometric function.
First, we are going to simplify the argument of the function.
22=22×2=12\dfrac{{\sqrt 2 }}{2} = \dfrac{{\sqrt 2 }}{{\sqrt 2 \times \sqrt 2 }} = \dfrac{1}{{\sqrt 2 }}
So, we have to evaluate cos⁡−1(12){\cos ^{ - 1}}\left( {\dfrac{1}{{\sqrt 2 }}} \right).
And, we know, cos⁡(π4)=12\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}
So, cos⁡−1(12)=π4{\cos ^{ - 1}}\left( {\dfrac{1}{{\sqrt 2 }}} \right) = \dfrac{\pi }{4}

Hence, the value of the given trigonometric function is π4\dfrac{\pi }{4}.

Note: In the given question, we had to find the value of the angle of a given trigonometric expression. This expression had just one basic trigonometric function which was equal to a value. But some students make mistakes when writing the angle in the form of pi. They do not know the value of the constant in terms of angle or forget to divide the conversion factor while changing the angle in degrees to the angle in pi. So, it is important that we know the value of pi and do not forget to correctly change it into the other form.