Look out! We’re diving into a more expansive view of trig by taking a look at the unit circle.
Transcript: Intro to the Unit Circle 1
This is where trigonometry really gets interesting. So, first I’ll say that one of the far ranging patterns in mathematics, this is big picture. Is that once mathematicians had figured out how something works in a limited context, the next step is always to consider how the idea can be expanded to a broader context.
And that’s exactly what we’re gonna do in this video. One of the most elegant examples of this in all of mathematics, it’s really actually incredible in the greater view of mathematics–is simply expanding trigonometry from the limited SOHCAHTOA context, to the much more inclusive unit circle context. So that’s what we’re gonna talk about in this lesson.
Expansion of Trig: The Unit Circle Context
And so I’m assuming at this point you’re comfortable with SOHCAHTOA, and that’s going to be our jumping off point. So, far we have defined everything. In the previous videos we’ve just talked about SOHCAHTOA in right angles. That’s all we’ve talked about, SOHCAHTOA and right triangles. And of course, in a right triangle, the acute angles must be bigger than 0 degrees, and smaller than 90 degrees.
That’s the range for the angle. And that’s a quite limited range if you think about it, because angles can be much much bigger than that. In the real world there many angles that are outside of that range, so consider the following. First of all, there are certainly architectural features that can have obtuse angles.
There’s are all kinds of geometric shapes. And we can see in the real world obtuse angles, we could have an obtuse angle between three points on a map, that sort of thing.
Now think about it. I could turn around once, or if I’m feeling really ambitious, I could turn around three times.
Well if I turn around three times, how much angle have I turned through? I’ve turned through 1,080 degrees of angle, okay? That just something simple I can do with my body. Now, let’s think about something mechanical, say the tire of a car. So let’s say the tire of a car, every time it turns is another 360 degrees. And say I drive that car from Boston to San Francisco, how much angle is that tire go through, probably millions of degrees.
So in the real world, there’s all kinds of angles that are much, much bigger than 90 degrees, and that’s why we need a larger context than SOHCAHTOA. To allow for the expanded range of angles, mathematicians move everything about trigonometry to the unit circle. So let’s briefly review. What is the unit circle?
What Is the Unit Circle?
The unit circle is in the xy plane. And it is a circle with a radius of 1, and a center at the origin. And so this is a picture of the unit circle. There it is in the xy plane, in the Cartesian plane. The equation of the unit circle is x squared + y squared = 1. This is something we talked about in the unit on coordinate geometry.
So that’s the equation of the unit circle. Notice that because it has a radius of 1, it has a circumference of 2 pi and has an area of pi. Those are the basic geometric facts about the unit circle. We will move our SOHCAHTOA triangle inside the unit circle, so that the vertex of the angle that we care about, the vertex of that angle is at the origin and the hypotenuse is the radius.
SOHCAHTOA Triangle Inside the Unit Circle
So here’s the SOHCAHTOA triangle situated inside the unit circle. Let’s think about this. So the radius has a length of 1 and so that’s the hypotenuse, okay? The point where the radius intersects the unit circle, we’re gonna call that point x, y.
And that’s gonna be a very special point, the point where that radius intersects the unit circle. And notice that x would be the horizontal distance from the y-axis to the point. In other words, this x distance here from the y axis over to that point, that’s the distance x and that equals the adjacent. Similarly, the y coordinate is the height above the x axis.
And so that’s equal to the opposite. And so something really important is happening here, let’s take a closer look. So again that horizontal distance from the origin along that horizontal leg, that is the distance x. And the vertical distance along the vertical leg from the x axis up to the point, that’s y.
Well this is really interesting. Because now it’s very clear that sine, that’s opposite over adjacent, y over 1, opposite over hypotenuse, so that’s y over 1. So that is just y, because of course the hypotenuse is 1. Similarly, the cosine, adjacent over hypotenuse, x over 1 is x. Sine equals y, cosine equals x.
Sine and cosine equal, respectively, the x and y coordinates of the point where the radius intersects the unit circle. And this in fact is the unit circle definition of sine and cosine. So in other words, sine and cosine in this system are no longer defined merely in terms of opposite and adjacent, they’re defined in terms of the point where that radius intersects the unit circle.
And the x coordinate is the cosine, and the y coordinate is the sine. The brilliance of this new definition is that, while it is perfectly consistent with the SOHCAHTOA understanding within that range, it’s 100% equal to our SOHCAHTOA understanding if the angle happened to be between 0 and 90, this new definition allows for unlimited angle. For example, when the angle is 0, that makes no sense in a triangle.
We can’t have a triangle with an angle of 0, then it would be a flat thing. It wouldn’t really be a triangle. But we can talk about that in the unicircle definition, because the radius; think about a radius at angle 0. Where does it intersect the unicircle? It intersects it along the x-axis, the positive x-axis, at (1,0).
So that’s the point of intersection of the radius with the unit circle,(1, 0). Well what that means is that sine of 0 equals the y coordinate, 0. And cosine of 0 equals the x coordinate, 1. In accordance with the uni circle definition we can say sine of 0 is 0, cosine of 0 is 1. And so right there, that is the value we get from a new definition that would not be possible.
That’s totally impossible in the SOHCAHTOA system, but our new system allows us to assign values to those. Let’s continue around the unit circle. Let’s go to 90 degrees, and again in a right triangle we only have one 90 degree angle, we can’t have two 90 degree angles in a triangle. But let’s just look at this, at 90 degrees, we’d be along the positive y-axis, we’d be at the point (0,1).
And so of course what that means is sine of 90 degrees is the wide coordinate 1, cosine of 90 degrees, is 0. All right, now we’ll go all the way around to 180. So now we’ll be entirely outside a triangle. Now we’re getting into angles that would never be in a triangle, but that’s fine because we’re in a new definition.
If we started the positive x-axis and go on 180 degrees, then we’re at the negative x-axis. And we’re at the point (-1, 0). Of course what this means is that the sine of 180 degrees is 0. And the cosine of 180 degrees is -1, so that’s interesting. So now we’re getting 0 outputs, but we’re also getting negative outputs, from cosine.
Keep on going around the unit circle. We get to 270 degrees. And so that’s three-quarters of the way around the circle, starting at the positive x axis going across the first quadrant, second quadrant, third quadrant, and then we wind up at the negative y axis. We’re at the point (0,-1).
And so here, sine of 270 degrees equals -1, cosine of 270 degrees equals 0. So it turns out both sine and cosine can have negative outputs as we move into other quadrants. We’ll talk more about that in the next video. I just wanna point out here, that the angle could be 360 degrees or much greater.
We just wrap around the unit circle multiple times, and then see where we land. We could go negative, and negative angle would just mean that we’re going clockwise, instead of counterclockwise. Again, we go around and see where we land. So the angle could be anything on the number line, so now in terms of function, so let’s think about this in terms of functions for a minute.
Sine and cosine. When we’re in the SOHCAHTOA region, they had a range between 0 and 90 degrees, and they had a domain between 0 and 90 degrees. That was a very limited domain in terms of the unit circle. Now these two functions have domains of all real numbers, and that is a gigantic change. Now, we’re talking about functions, that are defined every point on the real axis.
So here’s a practice problem. Pause the video, and then we’ll talk about this.
Okay, so this is a problem that’s just good practice, with the values on the axes, all these angles are values either on the x axis or the y axis. And so it’s just a matter of figuring out that point, and then figuring the sine and the cosine.
So which of the following is equal in value to sine of -270 degrees? Hm, well let’s think about that. Negative, that means it’s a clockwise angle. We’re gonna start at the positive x-axis, and then we’re gonna go a three-quarter turn in a clockwise direction, and so that’s where we’ll wind up.
We’ll start and we’ll go three-quarters the way around the circle, and we’ll actually wind up on the positive y-axis. And so that point is (1,0) where the unit circle ends, that’s where the final radius arm intersects the unit circle at (1,0). And sine is the y coordinate of that point, and so the sine of 270 degrees is the same as sine as 90, it’s 1.
Because in fact, the angle 90 and the angle -270 aren’t exactly the same place. So they both have the same sine and so that equals 1. And so really the question is, which of those following have the value of 1? Well, what’s the sine of 180 degrees, well 180 degrees is along the negative x axis, the y value there is 0, so sine of 180 degrees is 0, that doesn’t work. What’s the sine of 270 degrees?
Well, positive 270 degrees, we start at the start at the positive x-axis, we wrap three-quarters of the way around the circle to the negative y-axis. That point would be (0,-1), the y-coordinate would be -1. And so the sin(270) = -1. So that doesn’t work. The sine of 360 is the same as the sine of (0,0) and 360 are in the same place, we’re intersecting the unit circle there at (1, 0) on the positive x-axis.
The y-coordinate is 0, so this equals 0, so this doesn’t work. Now the cosine of 180, all right, well, let’s think about this. 180, we’re on the negative x-axis. The cosine is the x-coordinate, so that point on the negative x-axis Is the point (-1, 0). And so the x coordinate, cosine is the x coordinate.
The x coordinate is -1. And so cosine of 180 degrees is -1. This doesn’t work. Now cosine of 360, remember 360 is the same as 0. So if 0 and 360 at the same place, it’s the point (1,0). The x coordinate is 1, and so this equals 1, and so this has the same value, and so e is the answer.
The unit circle allows us to define sine and cosine, for all possible angles. And once again this is absolutely remarkable, because it allows us to expand our understanding of how sine and cosine work, from a very limited context of inside a triangle, opening it up to all possible angles of all possible sides. The radius at an angle theta, intersects the unit circle at the point (x,y). And that point, the point of intersection of the radius with the unit circle, the coordinates of that point provide the definition for sine and cosine.
So sine(theta) = y coordinate of that point and cos(theta) = x coordinate of that point. And we’ll talk about more implications of this definition, in the next video.