Using Unit Circles to Relate Right Triangles to Sine & Cosine

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  • 0:05 Everything Is Connected
  • 0:54 Right Triangles and Trig
  • 2:38 Sine and Cosine
  • 3:33 Unit Circle Triangles
  • 4:50 Lesson Summary
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Lesson Transcript
Instructor: Jeff Calareso

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

The unit circle is a helpful tool for understanding trigonometric concepts. In this lesson, we'll look at right triangles on the unit circle to better grasp sine and cosine.

Everything Is Connected

Did you ever get a sense that everything is connected? It's like that butterfly effect, where the breeze started by a butterfly flapping its wings goes on to cause a distant hurricane. The sense that things are connected is everywhere.

Maybe you can connect any actor to Kevin Bacon in six degrees or less. Or maybe you only met your best friend because you both dressed as metal bikini Princess Leia from Return of the Jedi for the same sci-fi convention, which was especially random since your best friend is (a) a guy and (b) really doesn't have the body to pull off a metal bikini. Not that most people do.

But anyway, when we're in the world of advanced mathematics, it's not just a coincidence that things seem connected. They totally are! Let's see how.

Right Triangles and Trig

Right triangles - those triangles with one right angle. Everybody loves right triangles, right? What's so great about them? Among other things, they have unique trigonometric properties.

If we look at this angle, theta, below, we can label the sides in relation to theta. The side opposite theta is... wait for it... the opposite side. The side adjacent is - yep - the adjacent side. And then there's the hypotenuse, which is the longest side.

Sides in a right triangle can be labeled in relation to the angle theta.
labeled right triangle

The three main trigonometric functions are sine, cosine and tangent. The sine of theta is equal to the opposite over the hypotenuse. The cosine of theta is equal to the adjacent over the hypotenuse. Finally, the tangent of theta is equal to the opposite over the adjacent. We abbreviate this with the phrase SOH CAH TOA.

But did you know that right triangles are connected to circles? Well, not just any circle. I mean the unit circle.

The Unit Circle

The unit circle is one of the magical math tools that make your life way easier. What's especially great about the unit circle is how simple it is. It's just a circle with a radius of 1.

How is it useful? Let's draw it. Start with a basic x- and y-axis. Now add a circle with a center at the origin. Again, the circle's radius is 1.

Unit circle
unit circle

Not much magic here. But this hurricane of awesome is just getting started.

If our radius line is here (seen below), and we draw another line down to the x-axis here (also seen below), we get what? A right triangle.

Right triangle in a unit circle
unit circle with right triangle

We'll call the angle above theta. And the triangle's hypotenuse is 1. Now let's get back to our trig.

Sine and Cosine

What is the sine of theta again? Remember SOH CAH TOA. Sine is the opposite over hypotenuse. So, the sine of theta is the side over the hypotenuse. But wait - if the hypotenuse is 1, then the sine of theta is just whatever this length is. So, we can call this leg below sine.

unit circle showing sine leg

And what about cosine? Cosine is the adjacent side over the hypotenuse. So, it's this side over 1, or just this side:

unit circle showing cos leg

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