Trigonometry is the study of relationships between sides and angles of a triangle. It is used widely in different fields like astronomy, geography, navigation, engineering, etc.
The values of sine, cosine, etc., are mostly used in all the above-discussed fields. We will be discussing more cosine and mainly about how to find the value of cos 120o.
Let us consider a right triangle ABC. Here ∠ ACB is an acute angle. From the above right triangle, AC is the hypotenuse, BC is the adjacent side, and AB is the opposite side to angle C.
Let us now see the values of trigonometric ratios of cos 0°, 30°, 45°, 60°,90°.
Now, let us find out the value of cos 120°.
Let us discuss a small concept of about two important and useful rules, which makes it easy to find the value of cos 120 degrees.
It says that the angles which lie between 0°− 90° are said to be in the first quadrant, angles between 90°− 180° be in the second quadrant, angles between 180°− 270° be in the third quadrant, and angles between 270°− 360° be in the fourth quadrant. This is called the quadrant rule.
This rule mainly helps us to memorize the quadrants at which sinθ, cosθ, and tanθ are positive.
First quadrant: All the functions are positive. Quadrant denoted as A.
Second quadrant: Only the function sine is positive. Quadrant denoted As S.
Third quadrant: Only tangent function is positive. Quadrant denoted as T.
Fourth quadrant: Only cosine function is positive. Quadrant denoted as C.
If we carefully observe here, starting from 4th quadrant to 1st quadrant, it is in the path of CAST, where,
C = Cosine. A = All.
S = Sine. T = Tangent.
From the above figure, 120° lies between 90o and 180°.Since 120°> 90° and 120°< 180°, so we can express 120° as (90° + 30°) and (180° − 60°).
1. (90° + 30°)
Cos 120° = Cos (90° + 30°)
We know, Cos (90° + θ) = − Sin θ
Cos (90° + 30°) = − Sin 30°
We also know, Sin 30° = ⇒− Sin 30° =−1/2
Cos 120° = −1/2
2. (180° − 60°)
Cos 120° = Cos (180° − 60°)
We know, Cos (180° −θ) = − cos θ
Cos (180° − 60°) = − cos 60°
We also know, cos 60° = ⇒− cos 60° = −1/2
Cos 120° = −1/2
Hence, the value of Cos 120° =−1/2