The formula
θ = arcsin(x), arccos(x) or arctan(x)
Where it comes from
The inverse functions undo the trigonometric ones, but they cannot undo them completely: infinitely many angles share the same sine, so each inverse is restricted to a range where it returns exactly one answer. That restriction is a convention, not a property of the maths.
How to work it out by hand
- Check the value is in range: sine and cosine only accept −1 to 1
- Apply the inverse function
- The result is the principal value, in the function's own range
- Add 360° repeatedly, or use the symmetry, to get the others
What is worth knowing
The restricted range is where most mistakes come from. Arcsine returns −90° to 90°, arccosine 0° to 180°, and arctangent −90° to 90°. So if you know the sine is 0.5, your calculator says 30° — but 150° also has a sine of 0.5, and in a triangle problem it may well be the answer you want. This is exactly the ambiguous case in the law of sines. Arctangent has a related trap: it cannot tell which quadrant you are in, which is why programming languages provide atan2 with two arguments instead of one.