Value of Cos 120

Cos 120 degree is equal to minus half which can be written as (- ½ ) or (-0.5).  Cos is one of the functions of Trigonometry that deals with the relationship between the angles and sides of a right-angled triangle. So in short, we can say that measuring a triangle (specifically right-angled triangle) is trigonometry.  

Now, the question arises what is a right-angled Triangle? We know that a triangle is a closed figure with three sides and three angles. The triangle having one of its interior angles as 90 degrees and the other two angles less than 90 degrees is called a right-angled triangle. Some of the important functions of Trigonometry are Sin, Cos, and Tan. The use of these functions finds its applications in the field of astronomy, engineering, architectural design, and physics.  

Let us take a right-angled triangle, select one of the angles as () and name three sides as follow:

Hypotenuse: The longest side of the triangle which is opposite to 90 degrees.

Perpendicular (Opposite): It is the side opposite to the unknown angle and perpendicular to the base (that is, the angle between base and perpendicular is 90 degrees).

Base (Adjacent): It is the side on which triangle rests and it also contains both the angles (90 degrees and unknown angle 

Most of the trigonometry calculations are done by using the trigonometry ratios. There are 6 trigonometry ratios present in trigonometry. Every other important trigonometry formula is derived with the help of these ratios.  

Given below are the important ratios named as Sin, Cos, and Tan. Sin and cos are the fundamental or basic ratios whereas Tan, sec, cot, and sec are derived functions. Given below are the ratios of Sin, Cos, and Tan.

1. sine function (sin)

\[sin\theta =\frac{opposite}{hypotenuse}\]

2. cosine function (cos)

\[cos\theta =\frac{adjacent}{hypotenuse}\]

3. tangent function (tan)

\[Tan\theta =\frac{sin\theta }{cos\theta }\]

Quadrant and Cast Rule

As stated above, the angles other than the 90 degrees angle in a right-angled triangle are acute (i.e, less than 90 degrees). To find the value of functions for angles more than 90 degrees, we follow a set of rules known as cast rules. Let us take four axes, to divide 360 degrees into four quadrants. The angles are always measured anti-clockwise from the positive x-axis. The given diagram shows 30 degrees from the positive x-axis. ImagewillbeuploadedsoonImagewillbeuploadedsoonImage will be uploaded soon.

There are four quadrants as shown in the figure below. Each quadrant contains a range of angles: 

• First Quadrant: All the angles between 0⁰ and 90⁰ lie in the first quadrant. The value of all the functions (Sin, Tan, Cos) in this quadrant are positive. Shown as A (represents All) in the second diagram given below.  

• Second Quadrant: All the angles between 90⁰ and 180⁰ lie in the second quadrant. The value of only Sin is positive in this quadrant. Shown as S (represents Sin) in the second diagram given below.  

• Third Quadrant: All the angles between 180⁰ and 270⁰ lie in the third quadrant. The value of only Tan is positive in this quadrant. Shown as T (represents Tan) in the second diagram given below.  

• Fourth Quadrant: All the angles between 270⁰ and 360⁰ lie in the fourth quadrant. The value of only Cos is positive in this quadrant. Shown as C (represents Cos) in the second diagram given below. 

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Starting from the fourth quadrant we can say that quadrant follows the series of CAST where C is for Cos, A for all, S  stands for Sin, and T is for Tan. 

Related Angles

The value of sin, cos, and tan of some angles are equal to the values of sin, cos, and tan of other angles. Let us take an example of cos(-30⁰). Since 30 degrees lie in the first quadrant we can say that cos(-30⁰) is equal to cos(30⁰) because all the angles in the first quadrant are positive. Similarly, cos(390⁰) is also equal to cos(30⁰). The diagrams below show the shaded angle of sin, cos, and tan having the same magnitude. 

Fig 1: sin 30⁰ = 0.5

Fig 2: sin 150⁰ = 0.5

Fig 3: sin 210⁰ = -0.5

Fig 4: sin 330⁰ = -0.5

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Therefore these angles are called related angles. The image given below shows the value of all angles of cos.

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Cos 120

The value of Cos 120⁰ is - ½. It is the additive inverse of the value of cosine 60⁰ or cos 60⁰. Cosine is one of the basic trigonometric functions. It is expressed as a ratio of the base of a right-angled triangle to its hypotenuse. 

How to Find the Value of Cos 120?

The angles of a right-angled triangle are expressed in terms of multiples or sub-multiples of 180⁰, or π in radians. Hence, to find the value of cos 120⁰, we will have to express 120⁰ in terms of 180⁰ or 90⁰. 

Case 1: 

Let us express 120⁰ as (180 - 60)⁰  

cos 120⁰ = cos (180 - 60)⁰  

Since, cos (180⁰ - x) = - cos x.

Therefore, cos (180 - 60)⁰ = - cos 60⁰ 

=> cos 120⁰ = -½ 

Case 2: 

Let us express 120⁰ as (90 + 30)⁰

cos 120⁰ = cos (90 + 30)⁰

Since, cos (90⁰ + x) = - sin x.

Therefore, cos (90 + 30)⁰ = -sin 30⁰ 

=>  cos 120⁰= - ½ 

Solved Examples 

Question 1: Find the Exact Value of cos (- 390⁰). 

Solution:  We know cos(-x) = cos(x) 

therefore , cos (- 390⁰) = cos (390⁰). 

Since the angle, 390⁰ is greater than angle 360⁰, 

we find an angle t such that,

t = 390 - (360) = 30⁰

This means Cos of cos (390⁰) and cos ( 30⁰)  coterminal.

So,

Cos (390⁰) = Cos ( 30⁰)

We know that the value of Cos 30⁰  is equal to \[\sqrt{\frac{3}{2}}\]

The value of cos 120 is minus half which can be written as -1/2 or -0.5. The study of cos or cosine comes under trigonometry which is a branch of mathematics that is concerned with the study of particular functions of angles and their application to calculations. Trigonometry is a branch of mathematics that mainly deals with the relationships between the sidelines and the angles of triangles. Trigonometry was first discovered in the Hellenistic period that is the third century BC, trigonometry is considered an extremely important study as it can be applied to the study of geometry to astronomical studies. The Greeks mainly delved into the concept of calculation of chords, while the expert mathematicians in India delved into the study of trigonometric ratios which are also known as trigonometric functions such as sine. Trigonometry is an extremely important study as it is applied to various disciplines such as serving, celestial mechanics, navigation et cetera.

Fun Fact-

• The father of trigonometry is called Hipparchus, as he is credited with compiling the first trigonometric table. Hipparchus used these trigonometric tables to solve the problems in trigonometry and spherical trigonometry.

• As we all know that trigonometric ratios are mainly the ratios between edges of a right triangle. These ratios are expressed in terms of trigonometric functions as a B and C which are the sides of the triangle.

• Sine function also called sin is defined as the ratio of the side which is opposite the angle to the hypotenuse.  \[cos\theta =\frac{Adjacent}{Hypoteneuse}\]

• Cosine function also called cos is defined as the ratio of the adjacent side to the hypotenuse.

• \[cos\theta =\frac{Adjacent}{Hypoteneuse}\]

• The tangent function also called tan is defined as the ratio of the opposite leg to the adjusting leg of the triangle.  \[Tan\theta =\frac{sin\theta }{cos\theta }\]

• These are the three main trigonometric ratios that are extremely important in the study of trigonometry and this article mainly deals with cos and the value of course 120.

• The hypotenuse in the right triangle is known as the side opposite to the 90° angle. It is the longest side of the triangle.

• The adjacent side is the side that is adjacent to its angle.

• The opposite side is known as the side which is opposite to the adjacent angle.

• The basic terms for adjacent and opposite are perpendicular and base which is mainly used in practice.