Question

How do you find the exact value of ${\cos ^{ - 1}}\left( { - 1} \right)$?

Answer 1

When the inverse of any trigonometric function is asked, it simply means that you need to find x for which $\cos x = - 1$. Here the thing to remember is that you can find x in degrees but that is frowned upon. The preferred result is in radians or in terms of $\pi$ where $2\pi = 360^\circ$.

Complete step by step solution:
When approaching this question, one needs to keep in mind that what it requires is the inverse of cos and as such, we need to find a value which when put in cos, will give the result which is the input of the inverse function. To understand it better, let us refer to our question.
It says that we need to find the value of ${\cos ^{ - 1}}\left( { - 1} \right)$ so let us assume the value to be x. What this statement means is that $x = {\cos ^{ - 1}}\left( { - 1} \right)$ which can further be interpreted as $\cos x = - 1$.
Now, we know that the value of cos is -1 at $180^\circ$. We can leave it at that but it is not the universally accepted form of representation. We need to convert it to radians or in the form of $\pi$.
For this, we know that $360^\circ = 2\pi$ which means that $180^\circ = \pi$.

As such we can say that ${\cos ^{ - 1}}\left( { - 1} \right) = \pi$.

Note:
The above answer seems correct and it very well is but only in the range $\left[ {0,2\pi } \right]$. After that, when moving on to other ranges, it is not correct because $3\pi ,5\pi ,7\pi$ and so on have values of their ${\cos ^{ - 1}}$ as -1 too. As such the complete answer will be ${\cos ^{ - 1}}\left( { - 1} \right) = \pi + 2n\pi$ where n belongs to natural numbers.