Question

Find the value of ${\cos ^{ - 1}}\left( {\cos \dfrac{{2\pi }}{3}} \right) + {\sin ^{ - 1}}\left( {\sin \dfrac{{2\pi }}{3}} \right)$.

Answer 1

Here, in this particular question we have to find the values and so for this question we have to use the property which is called the property of the inverse functions. That is ${\sin ^{ - 1}}\left( {\sin x} \right) = x{\text{ if x}} \in \left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$ and one more is${\cos ^{ - 1}}\left( {\cos x} \right) = x{\text{ if x}} \in \left[ {0,\pi } \right]$. So by using these two we can find the values.

Formula used:
Property of inverse functions used in this question are-
${\sin ^{ - 1}}\left( {\sin x} \right) = x{\text{ if x}} \in \left[ { - \dfrac{\pi }{2},\dfrac{\pi }{2}} \right]$
${\cos ^{ - 1}}\left( {\cos x} \right) = x{\text{ if x}} \in \left[ {0,\pi } \right]$

Complete step-by-step answer:
Since we have the function
${\cos ^{ - 1}}\left( {\cos \dfrac{{2\pi }}{3}} \right) + {\sin ^{ - 1}}\left( {\sin \dfrac{{2\pi }}{3}} \right)$
Here, $\dfrac{{2\pi }}{3} \notin \left[ {\dfrac{{ - \pi }}{2},\dfrac{\pi }{2}} \right]$. So by this, we can write the above statement as
$\Rightarrow {\cos ^{ - 1}}\left( {\cos \dfrac{{2\pi }}{3}} \right) + {\sin ^{ - 1}}\left( {\sin \left( {\pi - \dfrac{\pi }{3}} \right)} \right)$
And as we know from the property of inverse function, we can write the new equation as
$\Rightarrow {\cos ^{ - 1}}\left( {\cos \dfrac{{2\pi }}{3}} \right) + {\sin ^{ - 1}}\left( {\sin \dfrac{\pi }{3}} \right)$
And hence it can be written as
$\Rightarrow \dfrac{{2\pi }}{3} + \dfrac{\pi }{3}$
Since the denominator is the same so on adding both the number, we get
$\Rightarrow \dfrac{{2\pi + \pi }}{3}$
And on solving the above equation, we get
$\Rightarrow \dfrac{{3\pi }}{3}$
And therefore the common term will be canceled out, we have
$\Rightarrow \pi$
Therefore, the value ${\cos ^{ - 1}}\left( {\cos \dfrac{{2\pi }}{3}} \right) + {\sin ^{ - 1}}\left( {\sin \dfrac{{2\pi }}{3}} \right)$ will be equal to $\pi$.

Additional information:
As we have seen that without formula and the identities we can find the values of such functions. So I will tell you how we can easily remember these important identities. The sine, tangent, and secant will have a positive derivative while we have inverse functions, and the rest will have negative functions. So by this trick, we can easily remember it.

Note: Most of the time the inverse trigonometric properties are used to find the angle from the trigonometric angles associated with it. Inverse trigonometric functions are widely used in engineering, navigation, physics, and geometry. So it becomes necessary for us to remember not only for the solution but also for the later use of it.