Arccosine (arccos)
Calculate arccosine – the angle whose cosine is x.
It is the inverse operation of cos: while cos computes a ratio from an angle, arccosine (arccos) returns the angle from a ratio. The input x is from the domain [−1, 1], the resulting angle is in the range [0°, 180°].
What is arccosine
Arccosine is the inverse function of cosine – from a cosine value it returns the angle. Written arccos x or cos⁻¹ x.
Domain and properties
The domain is [−1, 1], the range is [0°, 180°].
Notable values
arccos 1 = 0°, arccos 0.5 = 60°, arccos 0 = 90°, arccos(−1) = 180°
Where arccosine is used in practice
- The angle between two vectors from their dot product (cosine similarity).
- Machine learning – cosine similarity when comparing texts and data.
- Physics – angle of incidence and reflection.
Inverse functions
An inverse trigonometric function does the opposite of the original function – it returns the angle from a value. Because trigonometric functions are periodic (they take the same value for infinitely many angles), the range of the inverse must be restricted to a single interval so the result is unambiguous. To compute a side from an angle, use the right triangle solver.
Relationships between the functions
tan α = sin α / cos α • cot α = 1 / tan α • sin²α + cos²α = 1
Tip: switch between degrees and radians. 180° = π rad ≈ 3.14159 rad.