>> In studying trigonometry using the idea of radian measure, we get high milage out of considering the unit circle. Now the unit circle is a circle whose radius is 1. So here's a unit circle. Now with radian measure, we know that theta is equal to arc-length over radius. That is if we're talking about an angle theta, a central angle of theta, then that central angle is equal to arc-length over radius. But gee, if we're talking about a unit circle then the radius is 1 and so theta corresponds with arc-length. Now so this opening of theta can be considered to be the same value as we have of the length of this arc. And I'm gonna call the length of that arc "T" just for the time being. Alright, now consider the idea that we can take a real number line. And let say I have a real number line in my hand right here. And let's say that, you know, from our previous experience the positive values lie to the right and the negative values to the left. Now look at what I'm gonna do. I'm just gonna take that number line, I'm gonna twist it like this, and I'm gonna twist it like this, and I'm gonna put 0 on the number line right here on that unit circle. And now I'm gonna take these positive real numbers and I'm gonna wrap them around this unit circle in this direction. And I'm just gonna keep on wrapping. I have an infinitely long line here with real numbers and I'm just wrapping that line around this circle. Alright, in the negative direction as well, I'm gonna take these negative real numbers and I'm just kinda wrap them around this unit circle, see the wrap, wrap, wrap, warp. Okay. Now I've got a situation where I have a length of an arc and of course it corresponds with an angle here. But let's forget about the angle for the moment. Let's just think about arc-length. So we have an arc-length, a real number you see, and that real number is associated then with a point because once we consider the real number T and we travel along the circle, that length, you see, then we arrive at a point on the circle and that point has coordinates and the coordinates of that point are XY. Now let's associate this idea of arc-length and the coordinates of that point on the unit circle with trig relationships or trig functions. Now there are six trig functions sine, cosine, tangent, cotangent, secant, and cosecant. And let's define them this way. Now sine is abbreviated as a S-I-N, cosine is C-O-S, tangent is T-A-N, and so on. Alright, the tangent of that real number T is simply the second component in the coordinates of that point where we land on the unit circle. So the sine of T is Y, the cosine of T is X, the tangent of T is Y over X, and cotangent T is X over Y, secant of T is 1 over X, cosecant of T is 1 over Y, so all of these just involve the coordinates of that point that we talked about. And also we noticed that we have a reciprocal relationship between tangent and cotangent. We have a reciprocal relationship between cosine and secant. We have a reciprocal relationship between sine and cosecant. Okay, and that's kind of a handy idea as well. Okay, so now we can relate all of these back to the notion of angle. It would be okay. But we wanna emphasize the idea of a real number. You see, if taking these functions of real numbers and associate them with the coordinates of points that lie on the unit circle. A really interesting twist in this study of trigonometry.