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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

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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.

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** - 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.

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** 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.

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.

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

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.

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

That means that the point at the yellow dot where the radius hits the edge of the circle can be identified as (cos(theta), sin(theta)). So, our right triangle butterfly flapped its wings, making sine and cosine. And that wind traveled all the way to our unit circle, where the same sine and cosine reappear. Our hurricane of awesome is really getting going now.

Let's see this hurricane spin (see video starting at 03:32). No matter where we move our radius line, it still hits the edge of our unit circle at (cos(theta), sin(theta)). As it moves, the values of cosine theta and sine theta just change as theta changes.

Here, theta is 0:

This really isn't a triangle, but the principle is unchanged. Cosine theta is still the distance from the origin to the yellow dot, so cosine theta is 1. What about sine theta? Since this is like a triangle where the side opposite theta is 0, then sine theta is 0.

What if we move theta to 90 degrees? Now cosine theta is 0, and sine theta is 1.

As we move into quadrant II, cosine theta becomes negative. Why? Our *x* values are negative. Sine theta stays positive until we get to quadrants III and IV. But in quadrant IV, cosine becomes positive again. It's all based on our *x* and *y* values.

So, the wind from the right triangle is perpetually part of our unit circle hurricane. It may blow north, east, south or west, but everything is connected.

In summary, we managed to connect butterflies and hurricanes to Princess Leia and trigonometry. More importantly, we looked at the connections between right triangles, the unit circle and sine and cosine.

Right triangles are triangles with one right angle. In a right triangle, the sine of an angle is equal to the opposite side over the hypotenuse. The cosine is equal to the adjacent side over the hypotenuse.

On the unit circle, we can create a right triangle by adding a radius line and connecting it to the *x*-axis. Since the radius of a unit circle is one, the hypotenuse of these triangles is also one. That means that the sine of the angle formed by the radius and the *x*-axis is whatever this length is here:

And the cosine is whatever this length is here:

Following this lesson, you should be able to:

- Explain the usefulness of a unit circle
- Describe how to create a right triangle on a unit circle
- Identify sine, cosine and tangent using a unit circle and a right triangle

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Math 103: Precalculus12 chapters | 94 lessons | 10 flashcard sets

- Go to Functions

- Graphing Sine and Cosine 7:50
- Graphing Sine and Cosine Transformations 8:39
- Graphing the Tangent Function: Amplitude, Period, Phase Shift & Vertical Shift 9:42
- Unit Circle: Memorizing the First Quadrant 5:15
- Using Unit Circles to Relate Right Triangles to Sine & Cosine 5:46
- Law of Sines: Definition and Application 6:04
- Law of Cosines: Definition and Application 8:16
- The Double Angle Formula 9:44
- Converting Between Radians and Degrees 7:15
- How to Solve Trigonometric Equations for X 4:57
- List of the Basic Trig Identities 7:11
- Go to Trigonometry

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