What is arctangent
Arctangent (written arctan or tan⁻¹) is the inverse of tangent: it answers "what angle has this tangent?" Since tan(45°) = 1, arctan(1) = 45°. Unlike arcsine and arccosine, it accepts any real number. When the known ratio involves the adjacent side and the hypotenuse instead, reach for the arccosine calculator.
How to use this calculator
Enter any number — positive, negative or zero — and read the angle in degrees. The output ranges from −90° to 90°. To go the other way, use the tangent calculator.
- Zero gives 0°, since tan(0°) = 0.
- Negative inputs give negative angles: arctan(−1) = −45°.
- Very large inputs approach 90° but never reach it, because tangent grows without bound near a quarter turn.
If your calculator or programming language expects radians, multiply the degree answer by π ÷ 180 to convert it back.
Slope to angle
Arctangent converts a slope (rise ÷ run) into an angle of inclination, which makes it handy in geometry and engineering. It joins arcsine and arccosine as the three inverse trig functions.
A ramp that climbs 1 m over 1 m of run has a slope of 1, so its angle is arctan(1) = 45°. A gentle 5% road gradient is a slope of 0.05 and works out to roughly 2.9°. The same trick converts a screen's aspect ratio into a diagonal angle, a vector's components into its heading, or a regression line's gradient into a tilt in degrees — anywhere a ratio of two lengths needs to become an angle.
Frequently asked questions
- What is arctan(1)?
- arctan(1) = 45°, the principal angle whose tangent is 1.
- What is the output range of arctangent?
- The principal value of arctangent ranges from −90° to 90°.
- Can arctangent accept any number?
- Yes. Unlike arcsine and arccosine, arctangent is defined for every real number.
- How do I turn a slope into an angle?
- Take the arctangent of the slope (rise ÷ run) to get the angle of inclination in degrees.
- When should I use atan2 instead of plain arctangent?
- Use atan2(y, x) when you need the direction of a point in all four quadrants. Plain arctan only sees the ratio y/x, so a point at (−1, −1) and one at (1, 1) both give 45°; atan2 keeps the signs separate and returns the full −180° to 180° range.