Arctangent (arctan)
Calculate arctangent – the angle whose tangent is x.
It is the inverse operation of tan: while tan computes a ratio from an angle, arctangent (arctan) returns the angle from a ratio. The input x is from the domain ℝ (any number), the resulting angle is in the range (−90°, 90°).
What is arctangent
Arctangent is the inverse function of tangent – from a tangent value it returns the angle. Written arctan x or tan⁻¹ x. Unlike arcsin and arccos it is defined for all real numbers.
Domain and properties
The domain is all real numbers, the range is (−90°, 90°). For very large x it approaches 90°, for very negative x it approaches −90°.
Notable values
arctan 0 = 0°, arctan 1 = 45°, arctan √3 = 60°
Where arctangent is used in practice
- Finding a slope angle from a gradient (e.g. roof pitch from known rise and run).
- Programming – the atan2 function converts coordinates to an angle (graphics, games, robots).
- Image processing – detecting edge orientation.
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.