Question

How do you find the value of $\operatorname{cosec} \,\pi $?

Answer 1

Hint:In this question we need to find the value of $\operatorname{cosec} \,\pi $. This question is related to finding the value of trigonometric functions to a corresponding degree. To find the value of, we need to know the definition of $\operatorname{cosec} \,$trigonometric function. It is a periodic function with its period equal to$2\pi $.

Complete step by step solution:
 Let us try to find the value of$\operatorname{cosec} \,\pi $. To find the value of$\operatorname{cosec} \,\pi $, we need to know the definition of $\operatorname{cosec} \,$function.

Trigonometric functions are those functions which provide a relationship between angles of the right angle triangle and its sides. $\operatorname{cosec} \,$(Cosecant) is also a one of those trigonometric functions.

In terms of the side of the right angle triangle, it is defined by Hypotenuse upon the perpendicular.

Let $\theta $ be the angle of a right angle triangle then

$\operatorname{cosec} \,\theta \, = \,\dfrac{{Hypotenuse}}{{Perpendicular}}$

As we know that$\sin \theta \, = \,\dfrac{{perpendicular}}{{Hypotenuse}}$.

So, we can see that there is a relation between $\sin \theta $and $\operatorname{cosec} \,\theta
$trigonometric function. Both of them are reciprocal of each other which means that
$\operatorname{cosec} \,\theta \, = \,\dfrac{1}{{\sin \theta }}$

We use this above relation to find the value of $\operatorname{cosec} \,\pi $. To find the value $\sin \pi $and use the relation between $\sin \theta $and $\operatorname{cosec} \,\theta $trigonometric function.

As we know that $\sin 2\theta \, = \,2\sin \theta \cos \theta $

Here$2\theta \, = \,\pi \Rightarrow \,\,\theta = \dfrac{\pi }{2}$. Putting values of $\theta $in the formula we got
\[\sin \pi = 2\sin \dfrac{\pi }{2}\cos \dfrac{\pi }{2}\]

As we know the value of $\sin \dfrac{\pi }{2} = 1$ and$\cos \dfrac{\pi }{2} = 0$. We get the value of$\sin \pi \, = \,0$.

From relation $\operatorname{cosec} \,\theta \, = \,\dfrac{1}{{\sin \theta }}$ we have
$\operatorname{cosec} \,\pi \, = \,\dfrac{1}{{\sin \pi }}$

As we know the value of$\sin \pi \, = \,0$. Also we have that dividing any number by 0 is undefined. So the value of

$\operatorname{cosec} \,\pi \, = \,\dfrac{1}{0} = \infty = undefined$

Hence $\operatorname{cosec} \,\pi $ is undefined at$\pi $.

Note: In the question which asked to find the value of trigonometric values for some degree value. We must have to know the definition of the trigonometric function and its identities or relation with trigonometric functions. Trigonometric functions have applications in finding the large distances.