Question

How do you find the value of $\csc \left( {\sin \left( {\dfrac{2}{3}} \right)} \right)$ ?

Answer 1

Hint : The above question is trigonometric function over trigonometric function. In these types, first try to find the value of the inner function, or the function nearest to the angle and then the other function sequentially. In the given function csc simply means inverse of the sine function respectively.

Complete step-by-step answer :
As the function with the angle is sin function, we will first try to find the value of the angle when sin is $\dfrac{2}{3}$ .
Let us assume the angle to be x. Therefore $\sin x = \dfrac{2}{3}$ when x = 41.81 and x = 180-41.81 = 138.19 respectively.
Hence we have to check the value of $\csc \left( {41.81} \right)$ and $\csc \left( {138.19} \right)$ .
We know that $\csc \left( x \right) = \dfrac{1}{{\sin \left( x \right)}}$
Hence, $\csc \left( {41.81} \right) = 1.49$
$\csc \left( {41.81} \right) = \dfrac{1}{{\sin \left( {41.81} \right)}} = \dfrac{1}{{0.67}} = 1.49$
And
$\csc \left( {138.19} \right) = \csc \left( { - 41.18} \right) = \dfrac{1}{{\sin \left( { - 41.18} \right)}} = \dfrac{1}{{ - 0.67}} = - 1.49$
Hence we have obtained two different values of
$\csc \left( {\sin \left( {\dfrac{2}{3}} \right)} \right)$ which are 1.49 and -1.49 respectively.
Also, $\csc \left( {41.81} \right) = 1.49$ lies in the I quadrant and $\csc \left( { - 41.81} \right) = - 1.49$ lies in II quadrant respectively.

Note : Cosecant or csc is not commonly used and is the inverse of sine function. In this type of question you just have to keep on finding the values of the trigonometric function to get the final answer. There may be some times a need to use trigonometric identities as well for further simplification.