 Let's continue our review of trigonometric functions by looking at a few more functions. So again, we have our unit circle, circle of radius one. We have our angle of theta. And again, the idea is that the measure of this angle is going to be determined by the length, the distance that we walk along the unit circle to go from the positive x-axis to this point that marks the end point of one of the sides of the angle. And if I extend this line out, this radial line outward, and I draw the tangent line from one, zero, this point of the unit circle. So I'm going to draw the tangent line. I'm going to extend this line until the two meet at some point. That gives me a larger triangle. And you might remember from geometry that the tangent line is always going to be perpendicular to the circle at the point of tangency. So this is not just a larger triangle, but this is also a right triangle where I have my right angle right here. And so I'm going to extend those two lines out. And sine and cosine both correspond to sine and cosine both correspond to length here. So here's my sine corresponds to this length. My cosine corresponds to this length. What I want is a new function that's going to correspond to this length here, which is the length of the tangent line. Let's see. What's a good name for length of the tangent line? I need a name that reminds me that I'm looking at tangent, tangent. Oh, let's call that the tangent. So we'll call this length here the tangent. And then we have one other length here, the length of this line here. Now, this term is a little bit more obscure, but this is a line that is not tangent to the circle, but we'll actually cross the circle at a couple of points. And we call that a secant line. And so we'll call the length of this line here the secant line. And you might be wondering, well, there's a third side of the triangle. What about this side here? So we've named the secant. We've named the tangent. What about this side? We need a name for this side. And that's actually pretty easy. This side of the triangle is the radius of the circle and actually has length one. That's an important thing to keep in mind for a little bit later on. So let's figure out what those tangent and secant values, well, we can use similar triangles to find them. So remember that I have this triangle in here and I have my radius one. I have my x-coordinate corresponding to this horizontal distance. I have my y-coordinate corresponding to that vertical distance. And then the next thing I need the other side of this triangle, I have one line here. So that's going to be my tangent value. And I can set up a proportionality using similar triangles. So I'll go, here's my angle. Opposite over adjacent y over x is opposite t over adjacent side. Again, remember this side here is the radius of the circle that has length one. And remember that when we define sine and cosine, cosine was the horizontal distance. Cosine was the x-coordinate. Sine was the vertical distance, the y-coordinate. And that gives us our relationship. Sine theta over cosine theta is t, our tangent line over one, which is just tangent of theta. And so this allows us to say that tangent of theta is actually the ratio of the quotient of sine over cosine. Likewise, if I want to take a look at the secant line. So here's my secant line. So here I'm going to do hypotenuse over adjacent one over x in the small triangle is hypotenuse that secant theta over adjacent. That's the radius of the circle. That's one in the big triangle. And so I have one over x secant over one by similar triangles. And again, remember that our x-coordinate corresponds to the cosine value. So that tells me that secant is one over cosine theta. Now, if we take a look at these unit circles, we have two very important trigonometric identities that emerge out of them. So our first one, again, remember that a unit circle, we're going to have our vertical distance corresponding to sine, our horizontal distance corresponding to cosine. And because this forms a right triangle, I get to use the Pythagorean theorem. And so what does the Pythagorean theorem tell me? Well, cosine squared plus sine squared, the sum of the squares on the legs of the triangle is equal to the square on the hypotenuse. So the Pythagorean theorem gives me sine squared plus cosine squared equal to one. And this is a very important trigonometric identity. In many ways, this is the fundamental trigonometric identity. And we usually call this the Pythagorean identity. Likewise, if I take a look at this triangle where I have the secant and tangent. So if I take a look at the triangle again, it's a right triangle because the tangent line is perpendicular to the radius of the point of tangency. And I have side one, side tangent theta, hypotenuse secant theta. And so the Pythagorean theorem tells me that one squared plus tangent squared is going to equal secant squared. And there's my second Pythagorean identity. Now algebraically, we only really need this one and the definitions of tangent and secant. Because I can take this Pythagorean identity, this fundamental trigonometric relationship. If I divide everything by cosine squared theta, I have sine squared over cosine squared. That's tangent squared. I have cosine squared over cosine squared. That's one. I have one over cosine squared. That's secant squared. So if I know this identity, I can derive this identity using a little bit of algebra. And if you ever forget this identity, well, it's useful to draw the picture because this is a nice geometric reminder that tangent is a length. Secant is a length. On the other hand, algebraically, if you know this, you can derive the second identity without any difficulty.