Trigonometric Functions as Projections

Our first introduction to trigonometric functions was as the ratio of the sides of right triangles. While this has some immediate applications, trigonometry is useful for vastly more than solving triangles, and we need to take another look at how we think of these functions.

These web pages are designed to help you explore trigonometric functions a projections based on the unit circle. In order to do this we need to start with some definitions.

The Unit Circle

Consider a circle centred at the origin of the Cartesian plane and with radius 1.0. This is what we mean when we refer to the unit circle. It's a unit circle because it has a radius of one. (The word unit means one.)

Signed Angles

We will also be working with the concepts of positive and negative angles. If you haven't encountered negative angles before this may seem strange, but the idea is really quite simple:

• positive angles are angles measured in an anti-clockwise direction; and
• negative angles are angles measured in a clockwise direction.

(Why choose positive for anti-clockwise angles and negative for clockwise angles? Good question! In fact, it is entirely arbitrary, and it would work just as well the other way around. But it was decided long ago to do it that way, and so we do. You might just as well ask why clocks go that way anyway!)

In considering trigonometric functions as projections on the unit circle, we will be measuring all our angles from the positive x-axis. We will be considering a ray beginning at the origin (0,0), and the point P where it intersects the unit circle, as shown in the interactive to the right. In the interactives you will be able to drag any red dots (like point P here) with the mouse. This will allow you to explore trig functions of different angles.

Finally a note about the letter used to represent the angle. It's common practice in trigonometry and geometry to use Greek letters to represent angles, most commonly θ theta and φ phi, but sometimes other pronumerals are used such as \(t\) or \(x\).

The interactives allow you to choose to measure angles in radians rather than degrees. Students encountering these ideas for the first time should just ignore this option and work in degrees, but this may be helpful for students revisiting these ideas after learning about radian measure.

Now, let's start looking at the sine function.