Introduction

Frequently we are in the situation of having to determine precisely which angles satisfy a particular equation. Something like . We know that , meaning that is a solution of this equation, but is that the only solution or are there more?

Let’s look at the graph of the sine function.

Notice that the graph of and the graph of the constant function intersect many times, not just once. In fact, since sine is a periodic function, these graphs intersect infinitely many times. Each of these intersections represents a single solution of the equation . We need a process to identify and write down each of these solutions. Let’s start by looking at the unit circle. Remember that sine values correspond to the -coordinate of points on the unit circle. This equation is asking us to find all the points on the unit circle with a -coordinate of . You see that there are two locations on the unit circle with -coordinate equal to , one in the first quadrant and another in the second. As we mentioned earlier, the first quadrant angle is . The angle in the second quadrant has reference angle , which means the angle is . Those are the only two points on the circle with that -coordinate, but remember that there are many other angles which are coterminal with those. For instance:

The only solutions are the angles , , and all the angles coterminal with them. Since the sine function has period , that means any other solution has to be an integer multiple of away from one of these first two solutions. Putting that together, our solutions are:

The steps we’ve followed are summarized in the following.

Let’s try one a bit more complicated.