Now that you understand how the program works, let's look for some patterns between a triangle's angle and its side lengths.

Let's take that general idea a little further with a specific example.

Is this pattern true for only isosceles triangles? Let's try another type of triangle.

Interesting. It seems like the angle doesn't depend on the side lengths, but on the ratio of the sides (how big one side is compared to another).

This pattern holds for all side lengths--the angle depends only on how "flat" or "thin" the triangle is--not how large the triangle is.

This is the main idea behind the trig functions. But using "flat" or "thin" is a little vague, so let's add some actual numbers. 

Comparing side lengths and angles is the foundation of trigonometry. In fact each side comparison (or ratio) has a specific name. Read this article to learn more about the terms.

Quick recap: each pair of triangle sides has a trig ratio it corresponds to. This trig ratio corresponds to a specific angle size. The ratios are: 

The trig ratios can be used to calculate the angles and sides of triangles, but we are going to focus on other properties. To learn more about solving triangles with trig try: here for a review and here for practice.

But why are the trig functions connected to triangles at all?

 Notice that both sine and cosine involve dividing by the hypotenuse. Well what happens when the hypotenuse is set to 1 to make the division simpler?

Remember every angle has a unique trig function (and vice versa). So this triangle doesn't just make every angle, it also makes every trig function.

The set hypotenuse makes calculating sine and cosine much easier:

and

Lets try some practice problems. Remember inside a unit circle sin = opposite (the height of the triangle) and cos = adjacent (the base of the triangle).

The height and width of the triangle are the same as the coordinates of point D. Width = x-coordinate; Height = y-coordinate.

Be careful not to mix them up!

Now that we have another way of measuring sine and cosine, we no longer need the triangle and can simply use the coordinates of the point on the circle.

This means we can find the sine and cosine of many angles.

Here's an interesting idea--what happens to those coordinate values as the radius travels around the circle?

Now lets graph the height of the point as the radius moves around the unit circle.

Repeating the process for cosine creates a very similar graph, but slightly shifted over!

Now that you know how all these ideas connect, all you have to do is to practice applying them to problems. With this background and a little memorization, the trig functions become less scary and far more interesting. 

Interview with Gianna