Question

How do you find the exact value of \[{\cos ^{ - 1}}\left( { - \dfrac{1}{2}} \right)\]?

Answer 1

Hint: 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 \[{\cos ^{ - 1}}\left( { - \dfrac{1}{2}} \right)\]. 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, which is \[ - \dfrac{1}{2}\].
Now, we know \[\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}\].
But we want the negative value of this constant. We know, the negative value of cosine exists only in the second quadrant, hence,
\[\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}\]
Thus, \[{\cos ^{ - 1}}\left( { - \dfrac{1}{2}} \right) = \dfrac{{2\pi }}{3}\]
Hence, the value of the given trigonometric function is \[\dfrac{{2\pi }}{3}\].

Note: In the given question, we just needed to know what the negative sign over the given trigonometric function means. Then we just used the concept that we got (which is the inverse of a function) and we simply just found the answer of the question.