 One thing that's important about trigonometric functions that you don't see for many functions in like an algebraic setting is the idea of trigonometric functions are periodic. They have a period for which they repeat themselves. So in general, if we have a function y equals f of x and there's some number p so that f of x plus p equals f of x, we can call that the period of the function if we choose the smallest positive number p so this happens. And so if we focus on the trigonometric functions for a moment, focus on say like the unit circle, like so. I know it's not the best looking circle in the world but it works. I mean, if you think of like sign is just the y-coordinate of the unit circle and cosine is the x-coordinate. What happens when you go from zero to two pi is you're gonna transverse the entire circle. Then when you do it again, it repeats itself. And then you do it again, it repeats itself. If you do it again and again and again and again and again it just repeats itself over and over and over again. So thinking about the points one, zero, zero, one, negative one and zero and then zero and negative one. If you think about y, the y-coordinate, you're gonna go from zero up to one, back down to zero, down to negative one, back up to zero, then repeats itself, zero to one, one to zero, zero to negative one, negative one to zero and this is gonna go over and over and over and over again. Sine is gonna be a periodic function and in fact, sine is gonna be a two pi periodic function. That is the period of sine is gonna be two pi. Once you go along the x-axis, two pi units, sine will repeat itself. So if we think about the graph of our function here, the graph, if this is like our x-axis right here, you have zero, two pi, whatever the graph turns out to be, you're just gonna repeat it when you go from two pi to four pi. So when we graph trigonometric functions, oftentimes if we can graph one period, then we can actually graph all of it because it just repeats itself. It's like a rubber stamp that we just stamp and stamp and stamp and just do it over and over and over again if we can know what one period looks like. So sine is a two pi periodic function. Cosine by similar reasoning is also two pi periodic. Now secant and cosecant are the reciprocals of cosine and sine and because sine and cosine are two pi periodic, secant and cosecant will also be two pi periodic. They will repeat themselves every two pi units along the x-axis. Now tangent and cotangent are also periodic functions. Tangent, for example, is just sine over cosine and cotangent is cosine over sine. So when you take tangent, we'll just use tangent as an example, you end up with sine of x over cosine of x in which case, well, since sine repeats itself every two pi and cosine repeats itself every two pi, tangent will also repeat itself every two pi units. But remember the definition of period is the smallest positive integer, excuse me, it doesn't have to be an integer, the smallest positive number such that it repeats itself. And so while tangent and cotangent do repeat themselves every two pi, it turns out a smaller cycle is actually possible. Tangent and cotangent actually repeat themselves every pi, that is every half spin, it doesn't have to be a full rotation. And the reason for that is the following, when you look at the first quadrant right here, sine and cosine are both positive in the first quadrant and the second quadrant you're gonna get, so these ones were both positive, right? In the second quadrant, x is negative and sine is positive, the y-cord is positive. In the third quadrant, they're both negative, right? And in the fourth quadrant, you're gonna get that x is positive and y is negative right there. Tangent is the ratio of this. So when you look at the first quadrant, tangent is positive over positive. In the second quadrant, which admittedly I should say this positive or positive is gonna be positive. In the second quadrant, you're gonna get positive over negative, which actually gives you a net negative. In the third quadrant, you're gonna get a negative over a negative, which gives you a positive, it's double negative. And in the fourth quadrant, you're gonna end up with a negative over a positive. So you have a net negative. So when you go from quadrant one to quadrant two, it'll go from positive, negative, get all the numbers. In the third quadrant to the fourth quadrant, you're gonna get positive to negative. So it turns out, when you look at the sines right here, the first and second quadrant actually give you everything that happens on a tangent function. The third and fourth quadrant will actually repeat the tangent pattern because the sines match up. The reason that sine and cosine are two pi periodic is because the sines don't, this cyclic nature of sines doesn't repeat itself until you go all the way around it. So if like for sine, you have to do positive, positive, negative, negative, positive, positive, negative, negative. For cosine, you're gonna do positive, negative, negative, positive, positive, negative, positive. The pattern doesn't repeat itself until you go one way around. But for tangent, it actually repeats itself pretty quickly. Positive, negative, positive, negative, positive, negative, positive, negative. It's an alternating sequence there. Cotangent does the same thing. So when working with tangent and cotangent, it's important to remember that they are pi periodic. They repeat themselves after a half rotation. You don't need a full rotation.