Question

How do you evaluate $\arcsin \left( \sin \left( \dfrac{10\pi }{11} \right) \right)$?

Answer 1

Hint: If we want to solve the inverse trigonometric identity $\arcsin \left( \sin \left( \dfrac{10\pi }{11} \right) \right)$ then we have to use the formula \[\arcsin \left( \sin x \right)=x\]. Here \[x\] is the angle. Then we have to simplify the angle and reduce it such that the coefficient of \[\pi \]in the numerator is 1. Here the inverse trigonometric identity \[\arcsin x\]can also be written as \[{{\sin }^{-1}}x\]

Complete step by step solution:
We have our given identity that is $\arcsin \left( \sin \left( \dfrac{10\pi }{11} \right) \right).....\left( 1 \right)$.
We have to simplify the angle of the identity; we know that \[\sin \left( \dfrac{10\pi }{11} \right)=\sin \left( \pi -\dfrac{\pi }{11} \right)\] so, we have to write the identity \[\sin \left( \dfrac{10\pi }{11} \right)\]into the form \[\sin \left( \pi -\dfrac{\pi }{11} \right)\] such that, we get:
$\begin{align}
  & \Rightarrow \arcsin \left( \sin \left( \dfrac{10\pi }{11} \right) \right) \\
 & \Rightarrow \arcsin \left( \sin \left( \pi -\dfrac{\pi }{11} \right) \right).....\left( 2 \right) \\
\end{align}$
Now, we have the identity (2). The identities are converted in simplified form because it will be easier to solve. Also we know that\[\sin \left( \pi -\dfrac{\pi }{11} \right)\]will be equal to \[\sin \dfrac{\pi }{11}\]. This is because in the first and the second quadrant the sine is always positive.
$\begin{align}
  & \Rightarrow \arcsin \left( \sin \left( \pi -\dfrac{\pi }{11} \right) \right) \\
 & \Rightarrow \arcsin \left( \sin \left( \dfrac{\pi }{11} \right) \right).....\left( 3 \right) \\
\end{align}$
Since we have obtained the identity (3), now we have to use the formula \[\arcsin \left( \sin x \right)=x\]. Applying the formula in the above identity, we get:
$\begin{align}
  & \Rightarrow \arcsin \left( \sin \left( \dfrac{\pi }{11} \right) \right) \\
 & \Rightarrow \dfrac{\pi }{11}....\left( 4 \right) \\
\end{align}$

Now, we have obtained the solution to the problem in identity (4). The solution of the given identity is $\dfrac{\pi }{11}$.

Note: In trigonometry we use two notations for angles that are degrees and radians. We could use both the notations but radian is much easier to understand than degree notation because in case of complex problems radian notation is easier to solve. In radian notation \[\pi \]means \[180{}^\circ \]