Question

Find the principal value of \[{{\sin }^{-1}}\left( \dfrac{1}{\sqrt{2}} \right)\] .

Answer 1

Hint: These types of problems are in general quite simple in nature and are easy to solve. To solve questions like these, we need to have some basic as well as to some extent advanced knowledge of trigonometric equations, relations and general values. We can solve this problem effectively once we understand all the core concepts involved behind this problem. Here we first of all need to guess what value of sin $\theta$ will result in \[\dfrac{1}{\sqrt{2}}\] . Once we have found out this, our problem is almost done. Since in this problem, the value given inside the inverse function is positive, we can solve the problem more smoothly.

Complete step by step answer:
Now we start off with the solution to the given problem by trying to figure out for what angle the value of sin theta turns out to be \[\dfrac{1}{\sqrt{2}}\] . We can very easily find that, the value of \[\sin {{45}^{\circ }}\] is equal to \[\dfrac{1}{\sqrt{2}}\] and thus we replace \[\sin {{45}^{\circ }}\] in place of \[\dfrac{1}{\sqrt{2}}\] to get,
\[{{\sin }^{-1}}\left( \sin {{45}^{\circ }} \right)\] .
Now we can very easily find out the value of \[{{\sin }^{-1}}\left( \sin {{45}^{\circ }} \right)\] . We write that,
\[{{\sin }^{-1}}\left( \sin {{45}^{\circ }} \right)={{45}^{\circ }}\] . We now convert the answer we got in degrees to radians. We do it to get,
\[{{45}^{\circ }}=\dfrac{\pi }{4}\]

Thus our answer to the problem is \[\dfrac{\pi }{4}\] .

Note:These types of problems require a thorough understanding of trigonometry equations, relations and general values. We need to be very careful while we try to figure out what angle would represent the particular given value. In case the value inside the inverse function is negative, then we need to check for the appropriate quadrant where the value lies and then solve it accordingly.