Question

How do you find the period of $ \operatorname{cosec} $ graph?

Answer 1

Hint:In this question we need to find the period of $ \operatorname{cosec} $ function. By the period of a function we mean that this function repeats its values after every fixed interval. For example, the sine function has a period of $ 2\pi $ . To find the period of $ \operatorname{cosec} $ function we use the definition of cosec function and basic trigonometric identities of $ \operatorname{cosec} $ function.

Complete step by step solution:
Let us try to solve this question in which we need to find the period of $ \operatorname{cosec} $ function.

To solve this we need to use the definition period and cosec function. So here is the definition of period of a function.

A function $ f $ have period if it is periodic, by periodic it means
 $ f(x + m) = f(x) $ for every $ m > 0 $
And m will be the period of function $ f $ .

As we know that $ \operatorname{cosec} x = \dfrac{1}{{\sin x}} $ and also we know that period of $ \sin
x $ is a periodic function with period $ 2\pi $ because we have trigonometric identity $ \sin (x + 2\pi ) = \sin
x $ which satisfy the definition of period.
 $ \sin (x + 2\pi ) = \sin x $ and $ 2\pi > 0 $ $ eq(1) $

By using the above property of $ \operatorname{cosec} $ function, we can write $ eq(1) $ as
 $
\dfrac{1}{{\operatorname{cosec} (x + 2\pi )}} = \dfrac{1}{{\operatorname{cosec} x}} \\
\Rightarrow \operatorname{cosec} x = \operatorname{cosec} (x + 2\pi ) \\
 $

And since $ 2\pi > 0 $ so we have satisfied the definition of periodic function. So the period of
 $ \operatorname{cosec} $ function exists and it is $ 2\pi $ .

Note: Also for the question of finding the period of trigonometric functions, we can find the period from looking at their graph. To solve these questions need to know the definition of periodic function, period and trigonometric function identities. Similarly we can find the period of secant function from the cosine function. And from tangent function we can find the period of cotangent function.