Question

How do I find the value of $\sin \left( {\dfrac{{5\pi }}{4}} \right)$?

Answer 1

Hint: Here, we are required to find the value of $\sin \left( {\dfrac{{5\pi }}{4}} \right)$. Thus, we will use the quadrants and determine that the given angle is exactly in which quadrant. With the help of this, we will be able to further solve the question, and using the trigonometric table, we will be able to find the exact value of the given trigonometric function.

Complete step by step solution:
In order to find the value of $\sin \left( {\dfrac{{5\pi }}{4}} \right)$
First of all, we know that we can write this as:
$\sin \left( {\dfrac{{5\pi }}{4}} \right) = \sin \left( {\pi + \dfrac{\pi }{4}} \right)$
Now, using the quadrants, when we add any angle to $\pi $ radians or $180^\circ $, then we reach the third quadrant.
And, as we know, in the third quadrant, $\sin \theta $ is negative.
Therefore, we can write $\sin \left( {\dfrac{{5\pi }}{4}} \right)$ as:
$\sin \left( {\dfrac{{5\pi }}{4}} \right) = \sin \left( {\pi + \dfrac{\pi }{4}} \right) = - \sin \left( {\dfrac{\pi }{4}} \right)$
Using the trigonometric table, $\sin \left( {\dfrac{\pi }{4}} \right) = \sin 45^\circ = \dfrac{1}{{\sqrt 2 }}$
$ \Rightarrow \sin \left( {\dfrac{{5\pi }}{4}} \right) = - \dfrac{1}{{\sqrt 2 }}$
If rationalized,
$ \Rightarrow \sin \left( {\dfrac{{5\pi }}{4}} \right) = - \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }} = - \dfrac{{\sqrt 2 }}{2}$

Therefore, the exact value of $\sin \left( {\dfrac{{5\pi }}{4}} \right)$ is $ - \dfrac{{\sqrt 2 }}{2}$.

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.