Arccos(x), cos-1(x), inverse cosine function.
Arccos definition
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
(Here cos-1 x means the inverse cosine and does not mean cosine to the power of -1).
Example
arccos 1 = cos-1 1 = 0 rad = 0°
Graph of arccos
Arccos rules
| Rule name | Rule |
|---|---|
| Cosine of arccosine | cos( arccos x ) = x |
| Arccosine of cosine | arccos( cos x ) = x + 2kπ, when k∈ℤ ( |
k is integer) |
| Arccos of negative argument | arccos(-x) = π - arccos x = 180° - arccos x | | Complementary angles | arccos x = π/2 - arcsin x = 90° - arcsin x | | Arccos sum | arccos(α) + arccos(β) = arccos( αβ - √(1-α2)(1-β2) ) | | Arccos difference | arccos(α) - arccos(β) = arccos( αβ + √(1-α2)(1-β2) ) | | Arccos of sin of x | arccos( sin x ) = -x - (2k+0.5)π | | Sine of arccosine | | | Tangent of arccosine | | | Derivative of arccosine | | | Indefinite integral of arccosine | |
Arccos table
| x | arccos(x) (rad) | arccos(x) (°) |
| --- | --- | --- | | -1 | π | 180° | | -√3/2 | 5π/6 | 150° | | -√2/2 | 3π/4 | 135° | | -1/2 | 2π/3 | 120° | | 0 | π/2 | 90° | | 1/2 | π/3 | 60° | | √2/2 | π/4 | 45° | | √3/2 | π/6 | 30° | | 1 | 0 | 0° |