Definition: Inverse trigonometric functions take a ratio of sides and return the acute angle that produces it; they are written sin^-1, cos^-1, and tan^-1.
What do they do?
Sine turns an angle into a ratio. Inverse sine turns a ratio back into an angle. If sin A = 0.5, then A = sin-1(0.5) = 30 degrees. The notation does not mean a reciprocal.
How do you use them?
Compute the ratio from two known sides of a right triangle, then press the inverse key on your calculator, in degree mode. Choose the function by the sides you know: opposite and hypotenuse use sine, adjacent and hypotenuse use cosine, opposite and adjacent use tangent.
Example. A ramp is 5.2 m long and rises 0.9 m. The angle with the ground satisfies sin A = 0.9 / 5.2, about 0.1731. So A = sin-1(0.1731), about 9.97 degrees. A roof with a run of 12 ft and a rise of 5 ft has tan A = 5/12, so A = tan-1(0.4167), about 22.6 degrees.
What mistakes occur?
Reading sin-1 as 1 / sin, and leaving the calculator in radian mode. For a right triangle, the inputs of inverse sine and cosine must lie between 0 and 1, or no triangle exists.