Cosine Function

Cos function (or cosine function) in a triangle is the ratio of the adjacent side to that of the hypotenuse. The cosine function is one of the three main primary trigonometric functions and it is itself the complement of sine(co+sine).

Other Trigonometric Functions

Sine Function Tan Function
Cosec (Csc) Function Sec Function
Cot Function  

Cosine Definition

In a right-triangle, cos is defined as the ratio of the length of the adjacent side to that of the longest side i.e. the hypotenuse. Suppose a triangle ABC is taken with AB as the hypotenuse and α as the angle between hypotenuse and base.

Now, for this triangle,

cos α = Adjacent Side/Hypotenuse

Cosine Formula

From the definition of cos, it is now known that it is the adjacent side divided by the hypotenuse. Now, from the above diagram,

cos α = AC/AB

Or,

cos α = b/h

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 Quadrants

It is interesting to note that the value of cos changes according to the quadrants. In the above table, it can be seen that cos 120, 150 and 180 degrees have negative values while cos 0, 30, etc. have positive values. For cos, the value will be positive in the first and the fourth quadrant.

Degree Range Quadrant Cos Function Sign Cos Value Range
0 to 90 Degrees 1st Quadrant + (Positive) 0 < cos(x) < 1
90 to 180 Degrees 2nd Quadrant – (Negative) -1 < cos(x) < 0
180 to 270 Degrees 3rd Quadrant – (Negative) -1 < cos(x) < 0
270 to 360 Degrees 4th Quadrant + (Positive) 0 < cos(x) <10

Cos Graph

The cosine graph or the cos graph is an up-down graph just like the sine graph. The only difference between sine graph and cos graph is that sine graph starts from 0 while the cos graph starts from 90 (or π/2). The cos graph given below starts from 1 and falls till -1 and then starts rising again.

Arccos (Inverse Cosine)

The cos inverse function can be used to measure the angle of any right angled triangle if the ratio of the the adjacent side and hypotenuse is given. The inverse of sine is denoted as arccos or \(cos^{-1}\).

For an right triangle with sides 1, 2, and √3, the cos function can be used to measure the angle.

In this, the cos of angle A will be, cos(a)= adjacent/hypotenuse.

So, cos(a) = √3/2

Now, the angle “a” will be cos−1(√3/2)

Or, a = π/6 = 30°

Important Cos Identities

  • cos(x) + sin(x) = 1
  • cos θ = 1/sec θ
  • cos (−θ) = cos (θ)
  • arccos (cos (x)) = x + 2kπ  [where k=integer]
  • Cos (2x) = cos(x) − sin(x)
  • cos (θ) = sin (π/2 − θ)

Below, all the other trigonometric functions in terms of cos function are also give.

Other Trigonometric Functions in Terms of Sine

Trigonometric Functions Represented as Sine
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)

Cos Calculus

For cosine function f(x) = cos(x), the derivative and the integral will be given as:

  • Derivative of cos(x), f′ (x) = −sin (x)
  • Integral of cos(x), ∫f (x) dx = −sin(x) + C)      [where C is the constant of integration)

Law of Cosines in Trigonometry

The law of cosine or cosine rule in trigonometry is a relation between the side and the angles of a triangle. Suppose a triangle with sides a, b, c and with angles A, B, C are taken, the cosine rule will be as follows.

According to cos law, the side “c” will be:

c= a+ b− 2ab cos (C)