Question

How do you find the values of $\theta$ given $\csc \theta =2$?

Answer 1

We will look at the definition of the cosecant function. Then we will convert the given trigonometric function using its definition. Then we will see the definition of inverse trigonometric function. We will use the inverse trigonometric function to find the value of the angle. We will consider the periodicity of the function and find a general solution.

Complete answer:
The given equation is $\csc \theta =2$. The cosecant function is defined as $\csc \theta =\dfrac{\text{Hypotenuse}}{\text{Opposite}}$ in a right angled triangle. We already know that the sine function is defined as $\sin \theta =\dfrac{\text{Opposite}}{\text{Hypotenuse}}$. So, we have a relation between the cosecant function and the sine function. This relation is given as
$\csc \theta =\dfrac{1}{\sin \theta }$
So, the given equation can be written as the following,
$\begin{aligned} & \dfrac{1}{\sin \theta }=2 \\\ & \therefore \sin \theta =\dfrac{1}{2} \\\ \end{aligned}$
Let us look at the definition of inverse trigonometric function. The inverse trigonometric function is defined as the inverse function of a trigonometric function which is used to find the value of the angle from the given trigonometric ratio. So, using this concept, we have the following,
$\theta ={{\sin }^{-1}}\left( \dfrac{1}{2} \right)$
Now, we know that $\sin \dfrac{\pi }{6}=\dfrac{1}{2}$. The sine function is a periodic function with a period of $2\pi$ radians. Therefore, the solution for the given equation is $\theta =\dfrac{\pi }{6}+2\pi n$ where $n=0,1,2,\ldots$.

Note: We should be familiar with the trigonometric functions, and their relations with each other. The use of inverse trigonometric functions is needed in such types of questions where we have to find the value of the angle. We should know the values of the trigonometric functions for the standard angles.