Question

How do we find the exact values of the six trig functions of angle ${120^ \circ }$ ?

Answer 1

Hint:To find the value of all the trigonometric functions of angle ${120^ \circ }$ , first we have to know about all the quadrants, on which the particular function comes positive or negative. And then compare ${120^ \circ }$ with other angles. In this way, we can find the exact values of the trigonometric functions.

Step by step solution:-
Firstly, we will mention all the six trigonometric functions are sin, cos, tan, cosec, sec, cot .
And with the given angle, the trigonometric functions are $\sin {120^ \circ }$ , $\cos {120^ \circ }$ , $\tan {120^ \circ }$ , $\cos ec{120^ \circ }$ , $\sec {120^ \circ }$ and $\cot {120^ \circ }$ .
So, first we will find the exact values of $\sin {120^ \circ }$ :-
The $\sin {120^ \circ }$ value can be identified using other trigonometric angle such as 60, 180 degrees and so on. Let us consider the value 120 degrees in the Cartesian plane. We know that the Cartesian plane is divided into four quadrants. The value ${120^ \circ }$ falls on the second quadrants. As the value of sine function in the second quadrant takes the positive value, the value of $\sin {120^ \circ }$ should be a positive value.
We know that:
${180^ \circ } - {60^ \circ } = {120^ \circ }$
Also, we know that the trigonometric identity $\sin ({180^ \circ } - x) = \sin x$
Now, $\sin ({180^ \circ } - {120^ \circ }) = \sin {120^ \circ }$
Therefore, $\sin {120^ \circ } = \sin {60^ \circ }$
From the trigonometric table, use the value of $\sin {60^ \circ }$ which is equal to $\dfrac{{\sqrt 3 }}{2}$ .
Hence, the value of $\sin {120^ \circ }$ is $\dfrac{{\sqrt 3 }}{2}$ .
Similarly,
$
  \cos {120^ \circ } = \cos ({180^{^ \circ }} - {60^{^ \circ }}) \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \cos {60^{^ \circ }} = - \dfrac{1}{2} \\
 $
$
  \tan {120^{^ \circ }} = \tan ({180^{^ \circ }} - {60^{^ \circ }}) \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \tan {60^{^ \circ }} = - \sqrt 3 \\
$
$
  \cos ec{120^{^ \circ }} = \cos ec({180^{^ \circ }} - {60^{^ \circ }}) \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \cos ec{60^{^ \circ }} = \dfrac{2}{{\sqrt 3 }} \\
 $
$
  \sec {120^{^ \circ }} = \sec ({180^{^ \circ }} - {60^{^ \circ }}) \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \sec {60^{^ \circ }} = - 2 \\
$
$
  \cot {120^{^ \circ }} = \cot ({180^{^ \circ }} - {60^{^ \circ }}) \\
  \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \cot {60^{^ \circ }} = - \dfrac{1}{{\sqrt 3 }} \\
 $
Note:- Trigonometric functions are also known as a Circular Functions can be simply defined as the functions of an angle of a triangle. It implies that the connection between the points and sides of a triangle are given by these trig capacities. The fundamental mathematical capacities are sine, cosine, tangent, cotangent, secant and cosecant.