cos(x), cosine function.
- Definition of cosine
- Graph of cosine
- Cosine rules
- Inverse cosine function
- Cosine table
- Cosine calculator
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
Example
b = 3"
c = 5"
cos α = b / c = 3 / 5 = 0.6
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 = (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
Example
arccos 1 = cos-1 1 = 0 rad = 0°
See: Arccos function
Cosine table
| x (°) |
x (rad) |
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° | 0 | 1 |
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