Question

How do you evaluate $\tan \left( {\dfrac{{5\pi }}{4}} \right)$ ?

Answer 1

Hint: Try to write $\dfrac{{5\pi }}{4}$ in the simple way from angle $0$ to $\dfrac{\pi }{2}$ We have to find the value of $\tan \left( {\dfrac{{5\pi }}{4}} \right)$. Use the identity of the trigonometry, $\tan \left( {\pi + x} \right) = \tan x$ and use the quadrant rule to find whether it will have positive or negative signs. Find the value of $x$ to find the value of $\tan \left( {\dfrac{{5\pi }}{4}} \right)$ exactly.

Complete Step by Step Solution:
In the question, we are given that we have to find the value of $\tan \left( {\dfrac{{5\pi }}{4}} \right)$. Therefore, we will try to solve it by using the identity of the $\tan $ which is, $\tan \left( {\pi + x} \right) = \tan x$ and place the positive or negative sign before it by using the quadrant rule. So, we should know about quadrant rule –
Quadrant 1 – In this quadrant all trigonometric functions are positive. This quadrant lies in the angle less than ${90^ \circ }$ and more than 0
Quadrant 2 – In this quadrant, the trigonometric functions sin and cosec are positive while cos, sec, tan and cot functions are negative. This quadrant lies in the angle less than ${180^ \circ }$ and more than ${90^ \circ }$
Quadrant 3 – This quadrant has tan and cot functions as positive and cos, sin, sec and cosec functions negative. This quadrant lies in the angle less than ${270^ \circ }$ and more than ${180^ \circ }$
Quadrant 4 – cos and sec functions are positive in this quadrant and sin, cosec, tan and cot functions are negative in this quadrant. This quadrant lies in the angle less than ${360^ \circ }$ and more than ${180^ \circ }$ .
Now, using the identity, $\tan \left( {\pi + x} \right) = \tan x$ in the question, $\tan \left( {\dfrac{{5\pi }}{4}} \right)$ -
$ \Rightarrow \tan \left( {\dfrac{{5\pi }}{4}} \right) = \tan \left( {\pi + \dfrac{\pi }{4}} \right)$
By using the identity, we get –
$ \Rightarrow \tan \left( {\dfrac{\pi }{4}} \right)$
This value of $\tan $is in the third quadrant as the angle is greater than ${180^ \circ }$ , so, the value of $\tan \left( {\dfrac{\pi }{4}} \right)$ will be positive. We know that, $\tan \left( {\dfrac{\pi }{4}} \right)$ is equal to 1.
$\therefore \tan \left( {\dfrac{{5\pi }}{4}} \right) = 1$

Hence, the value of $\tan \left( {\dfrac{{5\pi }}{4}} \right)$ is equal to 1.

Note:
Don’t go for calculating the exact value by using any other identity as this method gives the most efficient value. Use the quadrant rule to find the value of any trigonometric function correctly as it tells us whether the value of that trigonometric function will be positive or negative.