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Question: How do we find the exact values of the six trig functions of angle \({120^ \circ }\) ?...

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

Explanation

Solution

To find the value of all the trigonometric functions of angle 120∘{120^ \circ } , first we have to know about all the quadrants, on which the particular function comes positive or negative. And then compare 120∘{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∘\sin {120^ \circ } , cos⁡120∘\cos {120^ \circ } , tan⁡120∘\tan {120^ \circ } , cos⁡ec120∘\cos ec{120^ \circ } , sec⁡120∘\sec {120^ \circ } and cot⁡120∘\cot {120^ \circ } .
So, first we will find the exact values of sin⁡120∘\sin {120^ \circ } :-
The sin⁡120∘\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∘{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∘\sin {120^ \circ } should be a positive value.
We know that:
180∘−60∘=120∘{180^ \circ } - {60^ \circ } = {120^ \circ }
Also, we know that the trigonometric identity sin⁡(180∘−x)=sin⁡x\sin ({180^ \circ } - x) = \sin x
Now, sin⁡(180∘−120∘)=sin⁡120∘\sin ({180^ \circ } - {120^ \circ }) = \sin {120^ \circ }
Therefore, sin⁡120∘=sin⁡60∘\sin {120^ \circ } = \sin {60^ \circ }
From the trigonometric table, use the value of sin⁡60∘\sin {60^ \circ } which is equal to 32\dfrac{{\sqrt 3 }}{2} .
Hence, the value of sin⁡120∘\sin {120^ \circ } is 32\dfrac{{\sqrt 3 }}{2} .
Similarly,
cos⁡120∘=cos⁡(180∘−60∘)                   =−cos⁡60∘=−12  \cos {120^ \circ } = \cos ({180^{^ \circ }} - {60^{^ \circ }}) \\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \cos {60^{^ \circ }} = - \dfrac{1}{2} \\\
tan⁡120∘=tan⁡(180∘−60∘)                   =−tan⁡60∘=−3  \tan {120^{^ \circ }} = \tan ({180^{^ \circ }} - {60^{^ \circ }}) \\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \tan {60^{^ \circ }} = - \sqrt 3 \\\
cos⁡ec120∘=cos⁡ec(180∘−60∘)                        =cos⁡ec60∘=23  \cos ec{120^{^ \circ }} = \cos ec({180^{^ \circ }} - {60^{^ \circ }}) \\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \cos ec{60^{^ \circ }} = \dfrac{2}{{\sqrt 3 }} \\\
sec⁡120∘=sec⁡(180∘−60∘)                   =−sec⁡60∘=−2  \sec {120^{^ \circ }} = \sec ({180^{^ \circ }} - {60^{^ \circ }}) \\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \sec {60^{^ \circ }} = - 2 \\\
cot⁡120∘=cot⁡(180∘−60∘)                   =−cot⁡60∘=−13  \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.