Question

How do you evaluate $\arccos ( - 1)$ without a calculator?

Answer 1

We have the $\arccos ( - 1)$ cosine of the term and we have to simplify it; we will use the domain and range of the cosine function and find the required solution.

Complete step-by-step solution:
We have the given trigonometric function given to us as: $\arccos ( - 1)$
The function $\arccos (x)$ is the inverse function to the $\cos (x)$ function, therefore in this question we have to find the angle which the function $\arccos ( - 1)$ will give to us.
To find the required solution, we first have to find out the domain and range of the function $\arccos (x)$
We know the restrictions to the function are as follows:
The domain of the function is $[ - 1,1]$ and the range of the function is $[0,\pi ]$
Therefore, the only angle which has a $\cos ( - 1) = \pi$ because we know that $\cos (0)$ is $1$, now since we have the function as negative $1$, it will be in the second quadrant and the opposite value of $1$.

Therefore $\arccos ( - 1) = \pi$

Note: Basic trigonometric formulas should be remembered to solve these types of sums.
The inverse trigonometric function of $\cos x$ which is $\arccos (x)$ used in this sum
For example, if $\cos x = a$ then $x = \arccos (x)$
In some questions the inverse function is written as: ${\cos ^{ - 1}}x$, which is the function as $\arccos (x)$. It does the same work as the function $\arccos (x)$ does.
And $x = \arccos (\cos (x))$ is a property of the inverse function.
There also exists inverse functions for the other trigonometric relations such as $\sin$ and $\tan$.
The inverse function is used to find the angle $x$ from the value of the trigonometric function.
It is to be remembered that whenever there is a question on trigonometric functions, the question should be shifted in the simpler form of $\cos$ and $\tan$.