Graphing Sine and Cosine

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  • 1:08 The Unit Circle
  • 2:01 The Sine Wave
  • 4:00 The Cosine Wave
  • 5:20 Period
  • 6:20 Amplitude
  • 7:00 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.

Sine and cosine aren't just for triangles; they can also be waves. See them graphed on the unit circle and then learn how that translates into the sine and cosine waves.

Do the Wave

There are all different kinds of waves: ocean waves, stadium waves, microwaves, queen waves. What is a wave?

Think about a stadium wave. This is when everyone in a stadium - sometimes tens of thousands of people - are so collectively enraptured with whatever game is happening that they decide to stop paying attention and do this weird rolling cheer. It seems to happen at baseball games quite a bit. Not that baseball is boring. I do seem to root for a lot of losing teams. Maybe that's it...

Anyway, what happens when you do the wave? You're sitting, then you stand and throw your arms in the air. Then you sit and wait. Then it happens all over again. If all goes as planned, there's a wave that moves around the stadium. A wave that moves in a circle? Or is it a circle that moves in a wave? Either way, this is very much like what happens with sine and cosine waves.

The Unit Circle

Let's start with our stadium. This is the unit circle. It's a circle with a radius of 1 and a center at the origin, or (0,0). Not a bad place to catch a game, right?

When we draw a radius line like below, then add a line back to the x-axis, we have a right triangle. If this is supposed to be a baseball diamond, we're doing it wrong. But anyway, since our radius is 1, the sine of the angle theta is the vertical length over 1, or just the vertical length. The cosine of theta is just the horizontal side over 1, so, again, just the horizontal length.

Unit circle with a right triangle.
unit circle with right triangle

So, the yellow point, where the radius hits the circle, is (cos theta, sin theta). That's the great thing about the unit circle. It takes sine and cosine and turns them into x and y coordinates.

The Sine Wave

So, that's our stadium, such as it is. Now, let's start the wave. I'm pretty sure the wave is usually started by people discussing trigonometry.

What we're going to graph is y = sin x. We'll use the unit circle as a point of reference. So, we're using sine as a function, unraveling it from the unit circle. To start, when x = 0, what is sine? On our unit circle, it's a flat line, so it's 0. Let's put (0,0) on a new graph (final graph seen below).

For our x-axis, we'll use radians. When we get to pi / 4, that's 45 degrees - what a pretty triangle! What is the sine of 45 degrees? About 0.7. So, our wave is extending up from (0,0) past 0.7. See how the sine line is growing on our unit circle? Now, what happens at 90 degrees, or pi / 2? Well, y = 1. It's also the highest the sine line gets on our unit circle.

We then move into quadrant II, and the y values start getting smaller. So, as we graph our wave, we start falling. In quadrant III, we dip below zero. So, what does our wave do? It dips below zero, too. The lowest it gets is at 270 degrees, which is 3pi / 2. So, that's the bottom of our wave.

As we get into quadrant IV on the unit circle, we start going back towards zero. So, our wave curves up.

By the time we complete one full circle, at 2pi, we've completed a full wave. If we keep going, well, the wave just keeps going. It peters out when the fans lose interest or the game gets exciting. As long as it's going, we call this the sine wave. Look how it just follows the unit circle, going up and down as the y values go up and down.

The sine wave
sine wave and graph

The Cosine Wave

Now, not every stadium wave is the same. Your major league baseball wave is very different from your minor league wave. Even stadiums of comparable size have different waves. The same is true with trig. There's sine, and then there's cosine. Let's look at y = cos x.

When x is 0 on our unit circle, what is cosine? 1 (final graph seen below). Remember, cosine is the adjacent side of our triangle. At 0 degrees, there isn't much of a triangle, but cosine 0 is still 1. So, instead of starting at (0,0) like the sine wave, the cosine wave starts at (1,0).

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