If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Unit circle

## Video transcript

what I have attempted to draw here is a unit a unit circle and the fact that I'm calling it a unit circle means it has a radius of one so this length from the center and I centered it at the origin this length from the center to any point on the circle is of length one so what would this coordinate be right over there right where it intersects along the x axis well it would be X would be one Y would be zero well with this coordinate be up here well we've gone one above the origin but we haven't moved to the left or the right so our x value is 0 or Y value is 1 what about back here well here our x value is negative 1 we've moved one to the left and we haven't moved up or down so our Y value is 0 and what about down here well our we've gone what we've gone a unit down or one below the origin but we haven't moved in the X Y Direction so our X is 0 and our Y is negative 1 now with that out of the way I'm going to draw an angle and then this the way I'm gonna draw this thing I'm gonna define a convention for positive angles I'm gonna say a positive angle well the terminal sucks like the initial side of an angle we're always going to do along the positive x axis so this is the you can kind of view it as a starting side of the angle the initial side of an angle and then to draw an angle a positive angle the terminal side we're going to go we're gonna we're gonna move in a counterclockwise direction so positive angle means we're going counter clockwise counter clockwise this is just the convention I'm going to use and it's also the convention that is typically used and so you can imagine a negative angle would move in a clockwise clockwise direction so let me draw a positive angle so a positive angle might look might look something like this this is the initial side and then from that I go in a counterclockwise direction until I get until I measure out the angle and then this is the terminal side so this is a positive angle theta and what I want to do is think about this point of intersection between the terminal side of this angle and my unit circle and let's just say it has the coordinates a comma B the x-value where it intersects is a the y-value it intersects is B and I'm also the whole point of what I'm doing here is I'm going to see how this unit circle might be able to help us extend our traditional definitions of trig functions and so what I want to do is I want to make this theta part of a right triangle so to make it part of a right triangle let me drop an altitude right over here and let me make it clear that this is a 90 degree angle so this this theta is part of this right triangle so let's see what we can figure out about the sides of this right triangle so the first question I have to ask you is what is the length of the hypotenuse what is the length of the hypotenuse of this right triangle that I have just constructed well this hypotenuse is just a radius of a unit circle the unit circle has a radius of 1 so the hypotenuse has length 1 now what is the length of this blue side right over here this blue side you could view this as the opposite side to the angle well this height is the exact same thing is the exact same thing as the y-coordinate of this point of intersection of so this height right over here is going to be equal to be this the y-coordinate right over here is B this height is equal to B now exact same logic what is the length of this base going to be the base just of just of the right triangle well this is going to be the x-coordinate of this point of intersection if you were to drop this down this is the point X is equal to a or this whole length between the origin and that is of length is of length a now that we have set that up what is what is the cosine and what is the cosine you same green what is the cosine of my angle going to be in terms of A's and B's and any other numbers that might show up well to think about that we just need our sohcahtoa definition that's the only one we have now we are actually in the process of extending it sohcahtoa definition of trig functions and the cop part is what helps us with cosine it tells us that the cosine of an angle is equal to the length of the adjacent side over the hypotenuse so what's this going to be the length of the adjacent side for this angle the adjacent side has length a so it's going to be equal to a over what's the length of the hypotenuse well that's just 1 so the cosine of theta is just equal to a let me write this down again so the cosine of theta is just equal to a it's equal to the x-coordinate of where this terminal side of the angle intersected the unit circle now let's think about the sine of theta sine of theta and I'm going to do it in let me see I'll do it in orange so what's the sine of theta going to be well we just have to look at the so part of our sohcahtoa definition it tells us that sine is opposite over hypotenuse well the opposite side here has length B and what the hypotenuse has length 1 so our sine of theta our sine of theta is equal to B so an interesting thing this coordinate this point where our terminal side of our angle intersected the unit circle that point a B we could also view this as a is the same thing as cosine of theta a is the same thing as cosine of theta and B is the same thing B is the same thing as sine of theta well that's interesting that was just we use our sohcahtoa definition now can we in some way use this to extend sohcahtoa csoka Toa has a problem it works out fine if our angle is greater than 0 degrees if we're dealing with degrees and if it's less than 90 degrees we can always we can always make it part of a right but sohcahtoa starts to break down as our angle is either zero or maybe even becomes negative or as our angle is 90 degrees or more you can't have a right triangle with two 90-degree angles and it starts to break down let me make this clear so sure that's so this is a right triangle so the angle is pretty large I could make the angle even larger and still have a right triangle even larger but I can I can never get quite to 90 degrees at 90 degrees it's not clear that I have a right triangle anymore it all seems to break down and especially the case what happens when I go beyond 90 degrees so let's go let's see if we can use what we set up here let's set up a new definition of our trig functions which is really an extension extinct an extension of sohcahtoa and it's consistent with sohcahtoa instead of defining cosine as oh if i have a right triangle and saying okay it's the adjacent over the hypotenuse sine is the opposite over the hypotenuse tangent is opposite over adjacent why don't I just say for any angle I can draw it in the unit circle using this convention that I just set up and let's just say that the cosine of our angle the cosine of our angle is equal to the x-coordinate where we intersect is equal to the x-coordinate coordinate where the terminal side of our angle intersects the unit circle where where angel I'll write the terminal side terminal side of angle side of angle intersects intersects the unit the unit circle and why don't we define sine of theta sine of theta to be equal to the y-coordinate where the terminal side of the angle intersects the unit circle so essentially for any angle this point is going to define cosine of theta and what sine of theta and so what would be a reasonable definition for tangent of theta well tangent of theta even with sohcahtoa could be defined as sine of theta over cosine of theta which in this case is just going to be the y-coordinate where we intersect the unit circle over the x-coordinate in the next few videos I'll show some examples where we use the unit circle definition to start evaluating some trig ratios