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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.

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*.)

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.

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