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 side lengths of a right triangle relative to ฮธ
- 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). For a positive angle measured counterclockwise, use atan2 or add 180ยฐ/360ยฐ as appropriate.
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) loses the sign information and collapses quadrants II and IV onto I and III.
What if the adjacent side is zero? The ratio is undefined and the angle is exactly 90ยฐ. The "From sides" mode will flag this; switch to atan2 mode and enter x = 0 with your y value.
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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