Arcsine (arcsin)
Calculate arcsine – the angle whose sine is x.
It is the inverse operation of sin: while sin computes a ratio from an angle, arcsine (arcsin) returns the angle from a ratio. The input x is from the domain [−1, 1], the resulting angle is in the range [−90°, 90°].
What is arcsine
Arcsine is the inverse function of sine – from a sine value it returns the angle. Written arcsin x or sin⁻¹ x. It answers the question “which angle has a sine equal to x?”.
Domain and properties
The domain is [−1, 1] (a sine cannot exceed 1), the range is [−90°, 90°]. Outside [−1, 1] the function is undefined.
Notable values
arcsin 0 = 0°, arcsin 0.5 = 30°, arcsin 1 = 90°
Where arcsine is used in practice
- Finding an angle from a ratio of sides – when you know the sides of a triangle and need the angle.
- Robotics – computing joint angles from a desired arm position.
- Graphics and animation – finding a rotation angle from a position.
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.