Cosine function

cos(x), cosine function.

Cosine definition

In a right triangle ABC the sine of α, sin(α) is defined as the ratio betwween the side adjacent to angle α and the side opposite to the right angle (hypotenuse):

cos α = b / c


b = 3"

c = 5"

cos α = b / c = 3 / 5 = 0.6

Graph of cosine


Cosine rules

Rule name Rule
Symmetry cos(-θ) = cos θ
Symmetry cos(90°- θ) = sin θ
Pythagorean identity sin2(α) + cos2(α) = 1
  cos θ = sin θ / tan θ
  cos θ = 1 / sec θ
Double angle cos 2θ = cos2 θ - sin2 θ
Angles sum cos(α+β) = cos α cos β - sin α sin β
Angles difference cos(α-β) = cos α cos β + sin α sin β
Sum to product cos α + cos β = 2 cos [(α+β)/2] cos [(α-β)/2]
Difference to product cos α - cos β = - 2 sin [(α+β)/2] sin [(α-β)/2]
Law of cosines  
Derivative cos' x = - sin x
Integral ∫ cos x dx = sin x + C
Euler's formula cos x = (eix + e-ix) / 2

Inverse cosine function

The arccosine of x is defined as the inverse cosine function of x when -1≤x≤1.

When the cosine of y is equal to x:

cos y = x

Then the arccosine of x is equal to the inverse cosine function of x, which is equal to y:

arccos x = cos-1 x = y


arccos 1 = cos-1 1 = 0 rad = 0°

See: Arccos function

Cosine table





cos x
180° π -1
150° 5π/6 -√3/2
135° 3π/4 -√2/2
120° 2π/3 -1/2
90° π/2 0
60° π/3 1/2
45° π/4 2/2
30° π/6 3/2
0 1

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