 Welcome back to our lecture series math 1060 trigonometry for students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Missildine. In lecture eight, we are going to talk about circles. In particular, we're going to talk about how we can define, once again, the six trigonometric ratios using circles this time, instead of triangles like we've done previously in chapter two. Some things to remember about circles. The general equation of a circle was given as x minus h squared plus y minus k squared is equal to r squared, where the point h comma k is equal to the center of the circle. And this number r is then the radius of the circle. Every point on the circle is exactly r units away in the plane from this center. So if you set the center to be zero, zero, then what you see is that this h and this k disappears in the formula. And it simplifies just to be x squared plus y squared equals r squared. So this would be a circle centered at the origin with a radius of r. Well, if you set r equal to one, you get the so-called unit circle. So this is the circle which is centered at the origin and its radius is equal to one. And that's why we call it the unit circle. The radius is one unit of distance. And so using the unit circle, we can actually describe the six trigonometric ratios in the following manner. So notice the picture we see here on the right of the screen. This is a diagram of the unit circle. You'll notice that the center of the circle is the origin. That is the point where the x-axis and the y-axis intersect itself, they intersect themselves. And this circle will also have a radius of one. So every radius of this circle will be exactly one unit away from the origin. So we have this unit circle. OK, so if we take an angle theta and we put it into standard position, remember what that means, standard position means that the initial side of the angle coincides with the positive x-axis. And then the terminal side will terminate, well, wherever it does. It depends on the angle theta, right? In this diagram, we have the terminal side terminating in the first quadrant, but of course, that's not necessary. So considering the terminal side, though, right, the terminal side will have a point of intersection with the unit circle, which that point you see right here, this is a point in the plane, and that point has an x and y-coordinate. So let's say the point of intersection between the terminal side of the angle theta and the unit circle is this point x comma y. Well, what does a coordinate system mean after all? Well, x, the x-coordinate, represents how far to the right or left of the y-axis are you. If you're to the right of the y-axis, that would be a positive x-coordinate. If you're to the left of the y-axis, that means you have a negative x-coordinate. So x is measuring how far to the right of the y-axis are you. The y-coordinate, on the other hand, is measuring how far above the x-axis are you to arrive upon this point right here. And again, going above the x-axis gives you a positive y-coordinate, going below the y-axis gives you a negative y-coordinate like so. And so given this point, if we associate the usual right triangle that we have with these angle diagrams, you'd get a right triangle look something like this, all right, for which this interior angle is then the theta in question. The adjacent side would be this distance x, and the opposite side would be this number y right here. Now, because we're in the unit circle, the hypotenuse of this right triangle would be a radius of the z-circle, the unit circle, in which case that would be a one. So if we apply the usual trigonometric identities to this configuration right here, well, we'd get for sine, sine we take opposite over hypotenuse. y divided by one, which itself is just a y. So with respect to the unit circle, sine of theta is equal to y. That is the y-coordinate of a point on the unit circle is the sine ratio of the corresponding angle. Similarly, if we looked at cosine, cosine is going to be adjacent over hypotenuse, which in this case would look like x divided by one. But if you divide by one, it really doesn't do much, it's just an x. So cosine of theta is equal to x in this situation. So when you pick an arbitrary point on the unit circle and you take the associated angle with said point, theta right here, then the coordinates are just going to be cosine of theta times sine, not times, but comma times theta. So the x-coordinate of a point on the unit circle is cosine of the associated angle. And the y-coordinate of the point on the unit circle is just going to be sine of the associated angle theta. So cosine is the x-coordinate and y is just sine. Sine theta is just the y-coordinate of that point on the unit circle. Then we define the remaining four trigonometric ratios in the same manner for which we define tangent to be sine theta over cosine theta, which if sine is y and cosine is x, we get the simpler expression y over x. So if you take the y-coordinate divided by the x-coordinate, that ratio gives you the tangent ratio right here. Which if you look at that, a y divided by x, that kind of feels a little bit like a slope, doesn't it? And it turns out that in fact, tangent theta gets its name because it's measuring the slope of a so-called tangent line, something we might talk about another time. But continuing on, cosecant is the reciprocal of sine, therefore cosecant is 1 over y, secant is the reciprocal of cosine. So with respect to the unit circle, secant is just 1 over the x-coordinate. And then likewise, cotangent is the reciprocal of tangent. So if tangent is y over x, cotangent will be x over y with respect to this identification. So with the unit circle in mind, cosine is the x-coordinate, sine is the y-coordinate, and the other trigonometric ratios are derived from the usual relationships that we get from sine and cosine. So that's the important thing I want you to remember, that when we define sine and cosine using circles, the unit circle specifically, x gives us cosine, the x-coordinate gives us cosine, and sine coincides with the y-coordinate. So as an example of that, suppose the point P, which is given as negative 0.737 for x, and 0.675 for y, this is a point on the unit circle. What we want to do is we want to find sine of theta, cosine of theta, and tangent of theta, where theta is the arc of the unit circle from 1, 0 to P. So remember our unit circle right here, if we take the positive x-axis, this point over here is going to be 1, 0, right? It's the unit circle, so you go one distance away from the origin right here, you're going to get exactly 1, 0. And so if we take the arc that goes from 1, 0 all the way over here to this point in the second quadrant, mind you, notice it's in the second quadrant because the x-coordinate is negative, the y-coordinate is positive, you get this point negative 0.737, 0.675, like so. And so we're thinking theta is the angle associated to this arc of the circle right here. Well, how do you find sine of theta? Well, sine of theta is just the y-coordinate, so it's going to be 0.675. If you're a point on the unit circle, that's all one has to do to find sine. To find cosine, what do you do? You just grab the x-coordinate of the point on the unit circle, which is going to be negative 0.737, you don't need the parenthesis there, negative 0.737. That gives us sine and cosine. Tangent, this is the hardest one here. Tangent is going to be the y-coordinate divided by the x-coordinate, so you're going to get 0.675 divided by negative 0.737. This is the usual identity of sine divided by cosine. Feel free to help yourself with the calculator if you need to do that. I mean, we could do this by hand, but the arithmetic can get a little bit tedious, but if you put these numbers into a calculator, you end up with negative 0.916, which would then be the tangent ratio associated to this angle theta. And we could do secant and cosecant and cotangent if we wanted to, but I think this example is sufficient to see that as long as we can get sine and cosine, we can do the other trig ratios. And on the unit circle, cosine and sine are exactly the x and y-coordinates.