cos(x), cosine function

Contents hide

Cosine definition
Graph of cosine
Cosine rules
Inverse cosine function
Cosine table

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

For example:

b = 5″
c = 10″
cos α = b / c = 5 / 10 = 0.5

Graph of cosine

TBD

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 = (e^ix + 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:

Cosine table


x(^o)	x (rad)	cos x
180^o	π	-1
150^o	5π/6	-√3/2
135^o	3π/4	-√2/2
120^o	2π/3	-1/2
90^o	π/2	0
60^o	π/3	1/2
45^o	π/4	√2/2
30^o	π/6	√3/2
0^o	0	1