Inverse Functions

To fully understand how to solve a trigonometric function, first we must consider inverse functions

When solving an equation like x² = 9, we take the square root and say sqrt(9) = 3 using a calculator. Thus we say that taking the square root is the inverse of squaring. There is however, one small problem: the equation x² = 9 has two solutions 3 and -3, the minus sign disappears on squaring.

A calculator can only give one solution, and it chooses the positive one (the principal value). It is therefore important to understand the function to be able to solve equations involving inverses. Absolutely the best way of understanding functions is to be able to draw a sketch of the graph.

Now suppose we need to solve the equation sin x = 0.7. We use the inverse function sin^-1 to get x = 0.7754 radians. This is the principal value, but there are other solutions, e.g. x = 7.0585, x = -3.91699 (look at the sketch of the sin x). In fact there are infinitely many solutions.