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What is the Cosine Function?

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07 September 2020                

Read time: 6 minutes

Introduction 

You must have heard of a branch of mathematics known as Trigonometry.

This blog is on one of the widely used trigonometric functions named Cosine Function. 

In mathematics, the Trigonometric Functions are real-valued functions that relate an angle of a right-angled triangle to ratios of two sides of the triangle.

These functions are widely used in geometry and all sciences such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.

Cosine Introduction Image

The six trigonometric functions are the sine, the cosine, the tangent, the cosecant, the secant, and the cotangent. 

Also read:

All the trigonometric functions can be calculated from a right-angled triangle. 

  • "Opposite" is opposite to the angle, represented by a
  • "Adjacent" is adjacent (next to) to the angle, represented by b
  • "Hypotenuse" is the longest side, represented by h.

What is Cosine Function?

The cosine function is one of the primary mathematical trigonometric functions.

It is complementary to the sine function (co+sine).

Cosine function was first used in ancient Egypt in the book of Ahmes (c. 2000 B.C.). Much later F. Viète (1590) evaluated some values, E. Gunter (1636) introduced the notation "Cosi" and the word "cosinus" (replacing "complementi sinus"), and I. Newton (1658, 1665) found the series expansion for cosine.

The classical definition of the cosine function for the real angle is: "the cosine of an angle  in a right‐angle triangle is the ratio of the length of the adjacent side to the length of the hypotenuse."

Cosine Function triangle

Here, cos theta = adjacent/hypotenuse

This description of cos is valid for 0 <   θ < pi/2 when the triangle is non-degenerate.

In a right triangle, we see that there are three variables: the measure of the angle, and the lengths of the two sides (Adjacent and Hypotenuse). So if we know any two of them, we can find the third variable by using the above formula.
 

Example 1

 PR = 26 cm and PQ = 30 cm, find theta?

cosine example 1

Solution: From the formula, we know that the cosine of an angle is the adjacent side divided by the hypotenuse.  So we can calculate Cos  = 26/30

Cosine of theta is 0.866

  

Example 2

Find angle A?

Cosine example 2

Solution: In the figure above, three sides are given as AB = 3 cm, BC = 1 cm and AC = 2cm. In a right triangle, cos A  = adjacent/hypotenuse = AB/AC  = √3/2

cos A is √3/2 at  A = 30°

 

Example 3

In the figure, find x. 

cosine example 3

Solution: Using cos formula we get cos 30°  = adjacent/hypotenuse = x/15
                                                                       √3/2 = x/15
                     Solving this we get           x = (15) x (√3/2) = 7.5 x 1.73

                                                                                         = 12.975 units


x = 12.975 units


What are the basic properties of Cosine Function?

Important cos Identities

cos2 (x) + sin2 (x) = 1
cos (x) = 1/sec (x)
cos (−x) = cos (x)
arc cos (cos (x)) = x + 2kπ  [where k is an integer]
cos (2x) = cos2 (x) – sin2 (x)
cos (x) = sin (π/2 − x)

Example 4

If cos x is 34, find sin x and hence cos 2x. 

Solution: Using the identity,  cos2 (x) + sin2 (x) = 1
                                     We get, sin x  = √(1-cos2x)    = √(1-9/16)  = √7/4. 
                                                  cos2x = os2 (x) – sin2 (x)  = (3/4)2-(√7/4)2 = (9-7)/16  = 1/8

cos 2x = 1/8

   

All Trigonometric Functions in Terms of cosine
Any trigonometric function can be converted in terms of any other trigonometric function. Similarly, all trigonometric functions can be converted in terms of cosine function. The table below gives the formulas of five trigonometric functions in terms of cos function.

 Trigonometric Functions

 Represented as cosine

 sin θ

 ±√(1-cos2θ)

 tan θ

 ±√(1-cos2θ)/cos θ

 cot θ

 ±cos θ/√(1-cos2θ)

 sec θ 

 ±1/cos θ

 cosec θ

 ±1/√(1-cos2θ)

 

Cosine Table

 Cosine Degrees

 Values

 cos 0°

 1

 cos 30°

 √3/2

 cos 45°

 1/√2

 cos 60°

 1/2

 cos 90°

 0

 cos 120°

 -1/2

 cos 150°

 -√3/2

 cos 180°

 -1

 cos 270°

 0

 cos 360°

 1

Cosine Properties With Respect to the Trigonometry Quadrants

The value of all trigonometric functions changes in various quadrants. In the table above, we can observe that cos 120°, 150°, and 180° have negative values while cos 0°, 30°, etc. have positive values. For cos x, which depends on the sign of x, the value will be positive in the first and the fourth quadrants and negative in the second and the third quadrants.

trigonometry quadrants and cosine


How is the Cosine Graph?

The trigonometric ratios can also be considered as functions of a variable which is the measure of an angle.

The angle for all trigonometric functions may vary as any real number.

Using the values of cos at various angles in different quadrants, the graph of cosine function can be plotted and it comes out a continuous function as shown below. 

cosine graph

The cos graph comes out to be continuous, periodic, and symmetric about y-axis. 

Range, Domain of Cosine and other Properties

Using a unit circle, a circle with radius 1 unit, we can calculate the domain and range of cos(x). For any point on a unit circle, the cos(x) is equal to adjacent / 1.

unit circle and cosine

The measure of adjacent can be defined for all the points on the circle, indicating that the angle x can take any value, positive or negative.

So, the domain of cos(x) is all real numbers.

Also, the value of cos(x), depending on the point on the circle, can go to a maximum of 1 at x = 0֩ and a minimum of -1 at x = 180֩. So, the range of cos(x) is from -1 to 1.

The value of cos(x) changes periodically.

In short, for y = cos (x):

Domain of Cosine

 Domain of Cosine = (- ∞, + ∞) = Set of all Real numbers

Range of Cosine

Range of Cosine = [-1, +1]

Period of Cosine

2pi

x-intercepts

x = k pi/2, place k is an integer.

y-intercepts

y = 1

Symmetry

since cos (-x) = - cos (x) then cos (x) is an odd function and the graph of cosx is symmetric with respect to the origin.


Summary

Cosine function also known as cos x, cos theta, or cos function is basically one of the 6 trig functions.

Cos x is the ratio of the length of the opposite to the length of the adjacent.

It has various identities that help in inter-conversion of the rest of the ratios into cosine.

Cos theta values change their sign in the four trigonometry quadrants, the values are positive in first and fourth quadrants while negative in the other two.

The graph of cosine comes out to be smooth, periodic and symmetric. It can be used to derive various other properties of cosine functions. 

Written by Anupama Mahajan, Cuemath Teacher

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Cuemath, a student-friendly mathematics platform, conducts regular Online Live Classes for academics and skill-development and their Mental Math App, on both iOS and Android, is a one-stop solution for kids to develop multiple skills. Know more about the Cuemath fee here, Cuemath Fee


Frequently Asked Questions (FAQs)

Opposite of cos?

sec theta = 1/ cos theta

cos(-x)?

cos(-x) = -cos (x)

cos 0

The value of cos 0 degrees is 1

cos 30?

 The value of cos 30 degrees is √3/2

 cos 45?

The value of cos 45 degrees is 1/√2

 cos 60?

The value of cos 60 degrees is 1/2

cos 90?

The value of cos 90 degrees is 0

cos 180?

cos 180 is -1


External References 

Cosine explained Visually


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