Question

How do you find the value of $\csc \left( {\dfrac{\pi }{4}} \right)$?

Answer 1

Hint: Here, we are required to find the value of $\csc \left( {\dfrac{\pi }{4}} \right)$. Thus, we will use the relationship between various trigonometric functions, and with the help of that, we will be able to convert the given trigonometric function into sine whose value can be found using the trigonometric table. Substituting that value and solving the question further, we will be able to find the required value of $\csc \left( {\dfrac{\pi }{4}} \right)$. Thus, by this, we will be able to find the required answer.

Complete step by step solution:
In order to find the value of $\csc \left( {\dfrac{\pi }{4}} \right)$, we will use the relationship between trigonometric functions to find the required value.
As we know, $\csc \theta = \dfrac{1}{{\sin \theta }}$
Hence, substituting $\theta = \dfrac{\pi }{4}$, we get,
$\csc \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sin \left( {\dfrac{\pi }{4}} \right)}}$
As we know, using trigonometric table, $\sin \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\sqrt 2 }}$
Therefore, we get,
$\csc \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\dfrac{1}{{\sqrt 2 }}}} = \sqrt 2 $

Hence, the required value of $\csc \left( {\dfrac{\pi }{4}} \right)$ is $\sqrt 2 $
Thus, this is the required answer.

Note:
In this question, we have used trigonometry. Trigonometry is a branch of mathematics that helps us to study the relationship between the sides and the angles of a triangle. In practical life, trigonometry is used by cartographers (to make maps). It is also used by the aviation and naval industries. In fact, trigonometry is even used by Astronomers to find the distance between two stars. Hence, it has an important role to play in everyday life. The three most common trigonometric functions are the tangent function, the sine, and the cosine function. In simple terms, they are written as ‘sin’, ‘cos’, and ‘tan’. Hence, trigonometry is not just a chapter to study, in fact, it is being used in everyday life.