Question

How do you evaluate $\arcsin \left( \cos \left( 10\dfrac{\pi }{9} \right) \right)$?

Answer 1

To solve the given inverse trigonometric identity, $\arcsin \left( \cos \left( 10\dfrac{\pi }{9} \right) \right)$ we need to first convert the cosine into sine by using the formula, $\cos \left( \dfrac{\pi }{2}-x \right)=\sin x$ .We know that ${{\sin }^{-1}}(\sin x)=x$ provided that $-\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2}$ . Since we have known the domain and range check if $x$ lies in that range. Now evaluate to simplify $x$ to get the result.

Complete step by step answer:
The given inverse trigonometric identity is, $\arcsin \left( \cos \left( 10\dfrac{\pi }{9} \right) \right)$
Firstly, we need to convert the cosine into sine by using the formula, $\cos \left( \dfrac{\pi }{2}-x \right)=\sin x$
Upon converting the trigonometric function, we get,
$\Rightarrow \arcsin \left( \cos \left( 10\dfrac{\pi }{9} \right) \right)=\arcsin \left( \sin \left( \dfrac{\pi }{2}-\dfrac{10\pi }{9} \right) \right)$
Now evaluate the terms inside the sine function for easier evaluation.
$\Rightarrow \arcsin \left( \sin \left( \dfrac{9\pi }{18}-\dfrac{20\pi }{18} \right) \right)$
$\Rightarrow \arcsin \left( \sin \left( -\dfrac{11\pi }{18} \right) \right)$
Now we know that ${{\sin }^{-1}}(\sin x)=x$ provided that $-\dfrac{\pi }{2}\le x\le \dfrac{\pi }{2}$ .
Now here for us $x=\dfrac{-11\pi }{18}$
Let us check if it lies in that range.
$x=-1.91$ and is restricted in that domain.
So, we find the reference angle.
The reference angle will be,
$\Rightarrow \sin \left( -\dfrac{11\pi }{18}+\pi \right)$
$\Rightarrow \sin \left( \dfrac{-7\pi }{18} \right)$
Now this lies in the domain of the inverse sine function.
Upon using the inverse function formula, we get,
$\Rightarrow \arcsin \left( \sin \left( -\dfrac{7\pi }{18} \right) \right)=-\dfrac{7\pi }{18}$

Hence, $\arcsin \left( \sin \left( -\dfrac{7\pi }{18} \right) \right)$ will now be equal to $\dfrac{-7\pi }{18}$

Note: The inverse functions in trigonometry are also known as arc functions or anti trigonometric functions. They are majorly known as arc functions because they are most used to find the length of the arc needed to get the given or specified value. We can convert a function into an inverse function and vice versa.
Check where the trigonometric functions become negative or positive. Also, whenever the value is out of range or domain check the function’s periodicity and then subtract or add it with the general period to get it back into the range. Always check when the trigonometric functions are given in degrees or radians.