the inverse trigonometric functions

The ordinary trig functions take an angle and return a ratio: sin 30 degrees = 0.5. Very often you need to run that backwards — you know the ratio and want the angle. The inverse trigonometric functions do exactly that: arcsin(0.5) hands you back the angle 30 degrees. They are how you recover an angle from the sides of a triangle.

Written arcsin x (or sin^-1 x), arccos x, and arctan x, they answer 'which angle has this sine / cosine / tangent?' But there is a catch that gives them their character. Because sine repeats and takes the value 0.5 at infinitely many angles (30 degrees, 150 degrees, 390 degrees, and so on), the inverse cannot return all of them and still be a function. So each inverse is forced to choose ONE answer from a restricted range, called the principal value: arcsin returns angles in -90 to 90 degrees, arccos returns 0 to 180 degrees, and arctan returns -90 to 90 degrees. Thus arcsin(0.5) = 30 degrees, never 150, even though sin 150 degrees is also 0.5. Note the warning sign sin^-1 x means the inverse function, NOT 1/sin x, which is csc x — a notorious clash of notation.

These functions are essential whenever you solve a triangle for an unknown angle, find the angle of elevation from measured distances, or solve a trig equation. The recurring trap is the restricted range: your calculator gives only the principal value, so when a real problem could have an angle outside that range (an obtuse angle in a triangle, say), you must add the other solutions yourself using reference angles and periodicity.