Question

Find the value of \[{\cos ^{ - 1}}(\cos 13)\]

Answer 1

Hint: We use the concept that the inverse of a function cancels the same function when applied on the function and solves for the value of the given equation.
* If we have a function \[f(x)\]which is dependent on the independent variable ‘x’, then if the inverse of the function exists we can write \[{f^{ - 1}}(f(x)) = x\].
* Inverse of a function is that function that reverses the function from one form to another form by shifting the places of dependent and independent variables with each other. If we have a function \[y = f(x)\] which is dependent on the independent variable ‘x’ , then its inverse function will be denoted by \[{f^{ - 1}}\] and can be written as \[{f^{ - 1}}(y) = x\]

Complete answer:
We have to find the value of \[{\cos ^{ - 1}}(\cos 13)\]
Here we have the function as a trigonometric function.
Function given in the question is a cosine function.
The inverse of the function cosine cancels out if there is cosine function and the inverse is also of cosine function.
We solve the given equation where the function is also cosine and the inverse function is also cosine.
\[ \Rightarrow {\cos ^{ - 1}}(\cos 13) = 13\]

\[\therefore \]The value of \[{\cos ^{ - 1}}(\cos 13)\]is 13.

Note:
Alternate method:
First we calculate the value of the function inside the bracket using a calculator and then apply the inverse function to that value obtained.
We have to calculate the value of \[{\cos ^{ - 1}}(\cos 13)\] … (1)
So we first calculate the value of \[(\cos 13)\]using calculator
\[ \Rightarrow (\cos 13) = 0.90744678145\] … (2)
Now we apply cosine inverse function to this obtained value. Substitute the value from equation (2) in equation (1)
\[ \Rightarrow {\cos ^{ - 1}}(\cos 13) = {\cos ^{ - 1}}(0.90744678145)\]
Again use scientific calculator to calculate the inverse value
\[ \Rightarrow {\cos ^{ - 1}}(\cos 13) = 13\]
\[\therefore \]The value of \[{\cos ^{ - 1}}(\cos 13)\] is 13.