Calculate the arctangent angle from x, opposite and adjacent sides, or x and y coordinates, with results in degrees and radians for trig problems.
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Arc Tan Formula
The calculator uses one of three formulas depending on the input mode:
- ฮธ โ the resulting angle, returned in both radians and degrees
- x โ any real number (the tangent value)
- opposite, adjacent โ positive side lengths of a right triangle relative to ฮธ, entered in the same unit
- atan2(y, x) โ two-argument arctangent that uses the signs of x and y to place the angle in the correct quadrant
Plain arctan only returns angles in (โ90ยฐ, 90ยฐ). If you need the full angle from a coordinate, use the atan2 mode. Convert between radians and degrees with ฮธยฐ = ฮธrad ร 180/ฯ.
Reference Tables
Common arctan values you can use to sanity check a result:
| x | arctan(x) in degrees | arctan(x) in radians |
|---|---|---|
| 0 | 0ยฐ | 0 |
| 1/โ3 โ 0.5774 | 30ยฐ | ฯ/6 |
| 1 | 45ยฐ | ฯ/4 |
| โ3 โ 1.7321 | 60ยฐ | ฯ/3 |
| 10 | โ 84.29ยฐ | โ 1.4711 |
| โ โ | 90ยฐ | ฯ/2 |
For atan2, the quadrant of (x, y) determines the sign and range of the output:
| Point (x, y) | Quadrant | atan2(y, x) |
|---|---|---|
| x > 0, y > 0 | I | 0ยฐ to 90ยฐ |
| x < 0, y > 0 | II | 90ยฐ to 180ยฐ |
| x < 0, y < 0 | III | โ180ยฐ to โ90ยฐ |
| x > 0, y < 0 | IV | โ90ยฐ to 0ยฐ |
Example
A ramp rises 3 ft over a horizontal run of 8 ft. What angle does it make with the ground?
- tan ฮธ = opposite / adjacent = 3 / 8 = 0.375
- ฮธ = arctan(0.375) โ 0.3587 rad
- ฮธ โ 20.56ยฐ
FAQ
Why does my calculator show a negative angle? Arctan of a negative number is negative. The function is odd: arctan(โx) = โarctan(x). The atan2 result is a signed direction from the positive x-axis. To express a negative atan2 direction in the range 0ยฐ to 360ยฐ, add 360ยฐ. Do not add 180ยฐ without knowing the quadrant.
When should I use atan2 instead of arctan? Whenever you have both an x and a y component (vectors, slopes between two points, navigation bearings). Plain arctan(y/x) cannot distinguish opposite directions: it maps quadrant II to quadrant IV, and quadrant III to quadrant I.
What if the adjacent side is zero? A right triangle requires both side lengths to be positive. For a vertical direction, choose Coordinates (atan2) and enter x = 0: positive y gives 90ยฐ, negative y gives โ90ยฐ, and (0, 0) has no direction.
Radians or degrees? Degrees are standard for geometry and construction. Radians are required for calculus and most programming languages. The result panel shows both.
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