What an inverse trig function gives you
sin, cos and tan take an angle and give a ratio. The inverse functions go the other way: give arctan a slope of 1 and it returns 45°, the angle whose tangent is 1. Use them to find an angle in a right triangle from two sides, the direction of a vector, a roof or ramp angle from its slope, or the phase of a complex number.
Principal values: why there's one answer
Infinitely many angles share the same sine (30°, 150°, 390° and so on), so each inverse function picks one range of angles, its principal values:
arccos x: x from −1 to 1, angle from 0° to 180° (0 to π)
arctan x: any x, angle strictly between −90° and 90°
arccsc x = arcsin(1/x), arcsec x = arccos(1/x), for |x| ≥ 1
arccot x: any x, angle strictly between 0° and 180° (see below)
atan2(y, x): the angle of the point (x, y), from just above −180° up to 180°
To get radians from degrees, multiply by π ÷ 180.
Worked example: what is arctan of 1?
You want the angle whose tangent is 1, which means opposite = adjacent. In a right triangle with two equal legs, the angles are 45° and 45°. So arctan(1) = 45° = π/4 ≈ 0.7853981634 radians. Similarly arctan(2) ≈ 63.4349°, or 1.107149 radians.
Exact values for standard angles
| x | arcsin x | arccos x |
|---|---|---|
| −1 | −π/2 (−90°) | π (180°) |
| −√3/2 | −π/3 (−60°) | 5π/6 (150°) |
| −√2/2 | −π/4 (−45°) | 3π/4 (135°) |
| −1/2 | −π/6 (−30°) | 2π/3 (120°) |
| 0 | 0 | π/2 (90°) |
| 1/2 | π/6 (30°) | π/3 (60°) |
| √2/2 | π/4 (45°) | π/4 (45°) |
| √3/2 | π/3 (60°) | π/6 (30°) |
| 1 | π/2 (90°) | 0 |
| x | arctan x |
|---|---|
| −√3 | −π/3 (−60°) |
| −1 | −π/4 (−45°) |
| −√3/3 | −π/6 (−30°) |
| 0 | 0 |
| √3/3 | π/6 (30°) |
| 1 | π/4 (45°) |
| √3 | π/3 (60°) |
You can type these values directly, for example sqrt(3)/2, and the calculator will recognize the exact angle.
atan2 and the quadrant problem
arctan(y ÷ x) can't tell (1, 1) from (−1, −1), because both give a ratio of 1. atan2(y, x) looks at the signs of x and y separately, so it returns the true direction of the point. atan2(1, −1) is 135° (3π/4), while arctan(1 ÷ −1) is −45°. It matches the principal argument of the complex number x + yi.
Limitations
- Domain errors. arcsin and arccos only accept −1 to 1, because sine and cosine never leave that range. arccsc and arcsec only accept values of at least 1 in size. Outside those ranges there's no real angle, and the calculator says so.
- arccot convention. This page uses the range 0° to 180°, which keeps arccot continuous. The NIST Digital Library of Mathematical Functions defines arccot x as arctan(1/x), which gives a negative angle for negative x. When x is negative, the page shows both.
- Answers are real angles only. Complex inverse functions aren't covered.
Sources
- OpenStax Precalculus 2e, 6.3: Inverse trigonometric functions (domains and ranges)
- NIST Digital Library of Mathematical Functions, §4.23: Inverse trigonometric functions (principal values; arccsc, arcsec, arccot)
- NIST Digital Library of Mathematical Functions, §1.9: phase (argument) of a complex number, principal value in (−π, π]