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

Learn how to use the unit circle to define sine, cosine, and tangent for all real numbers. Created by Sal Khan.

Want to join the conversation?

  • blobby green style avatar for user Vamsavardan Vemuru
    Do these ratios hold good only for unit circle? What if we were to take a circles of different radii?
    (176 votes)
    Default Khan Academy avatar avatar for user
  • piceratops ultimate style avatar for user Hemanth
    What is the terminal side of an angle?
    (111 votes)
    Default Khan Academy avatar avatar for user
  • orange juice squid orange style avatar for user Rory
    So how does tangent relate to unit circles? And what is its graph?
    (17 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user William Hunter
      I think the unit circle is a great way to show the tangent. While you are there you can also show the secant, cotangent and cosecant. I do not understand why Sal does not cover this.
      Using the unit circle diagram, draw a line “tangent” to the unit circle where the hypotenuse contacts the unit circle. This line is at right angles to the hypotenuse at the unit circle and touches the unit circle only at that point (the tangent point). Extend this tangent line to the x-axis. The distance of this line segment from its tangent point on the unit circle to the x-axis is the tangent (TAN). If you extend the tangent line to the y-axis, the distance of the line segment from the tangent point to the y-axis is the cotangent (COT). To determine the sign (+ or -) of the tangent and cotangent, multiply the length of the tangent by the signs of the x and y axis intercepts of that “tangent” line you drew. For example, If the line intersects the negative side of the x-axis and the positive side of the y-axis, you would multiply the length of the tangent line by (-1) for the x-axis and (+1) for the y-axis. You will find that the TAN and COT are positive in the first and third quadrants and negative in the second and fourth quadrants.
      As a bonus, the distance from the origin (point (0,0)) to where that tangent line intercepts the x-axis is the secant (SEC). The sign of that value equals the direction, positive or negative, along the x-axis you need to travel from the origin to that x-axis intercept. The distance from the origin to where that tangent line intercepts the y-axis is the cosecant (CSC). The sign of that value equals the direction positive or negative along the y-axis you need to travel from the origin to that y-axis intercept.
      Some people can visualize what happens to the tangent as the angle increases in value. When the angle is close to zero the tangent line is near vertical and the distance from the tangent point to the x-axis is very short. As the angle nears 90 degrees the tangent line becomes nearly horizontal and the distance from the tangent point to the x-axis becomes remarkably long. You can, with a little practice, “see” what happens to the tangent, cotangent, secant and cosecant values as the angle changes.
      The second bonus – the right triangle within the unit circle formed by the cosine leg, sine leg, and angle leg (value of 1) is similar to a second triangle formed by the angle leg (value of 1), the tangent leg, and the secant leg. In this second triangle the tangent leg is similar to the sin leg the angle leg is similar to the cosine leg and the secant leg (the hypotenuse of this triangle) is similar to the angle leg of the first triangle. When you compare the sine leg over the cosine leg of the first triangle with the similar sides of the other triangle, you will find that is equal to the tangent leg over the angle leg. Therefore, SIN/COS = TAN/1 . You can also see that 1/COS = SEC/1 and 1^2 + TAN^2 = SEC^2. A bunch of those almost impossible to remember identities become easier to remember when the TAN and SEC become legs of a triangle and not just some ratio of other functions. The angle line, COT line, and CSC line also forms a similar triangle.
      -------------------------------------
      When you graph the tangent function place the angle value on the x-axis and the value of the tangent on the y-axis. At the angle of 0 degrees the value of the tangent is 0. At 45 degrees the value is 1 and as the angle nears 90 degrees the tangent gets astronomically large. You are left with something that looks a little like the right half of an upright parabola. At negative 45 degrees the tangent is -1 and as the angle nears negative 90 degrees the tangent becomes an astronomically large negative value. This portion looks a little like the left half of an upside down parabola. This pattern repeats itself every 180 degrees. Do yourself a favor and plot it out manually at least once using points at every 10 degrees for 360 degrees. It may not be fun, but it will help lock it in your mind. Give yourself plenty of room on the y-axis as the tangent value rises quickly as it nears 90 degrees and jumps to large negative numbers just on the other side of 90 degrees.
      (121 votes)
  • blobby green style avatar for user Jason
    I hate to ask this, but why are we concerned about the height of b? What is a real life situation in which this is useful? Graphing sine waves?
    (18 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Aaron Sandlin
      Say you are standing at the end of a building's shadow and you want to know the height of the building. you only know the length (40ft) of its shadow and the angle (say 35 degrees) from you to its roof. you could use the tangent trig function (tan35 degrees = b/40ft)
      40ft * tan35 = b
      28ft = b
      Now you can use the Pythagorean theorem to find the hypotenuse if you need it.
      (30 votes)
  • leaf blue style avatar for user Ram kumar
    In the concept of trigononmetric functions, a point on the unit circle is defined as (cos0,sin0)[note - 0 is theta i.e angle from positive x-axis] as a substitute for (x,y). This is true only for first quadrant. how can anyone extend it to the other quadrants? i need a clear explanation... I think trigonometric functions has no reality( it is just an assumption trying to provide definition for periodic functions mathematically) in it unlike trigonometric ratios which defines relation of angle(between 0and 90) and the two sides of right triangle( it has reality as when one side is kept constant, the ratio of other two sides varies with the corresponding angle).... i think mathematics is concerned study of reality and not assumptions.... how can you say sin 135*, cos135*...(trigonometric ratio of obtuse angle) because trigonometric ratios are defined only between 0* and 90* beyond which there is no right triangle... i hope my doubt is understood..... if there is any real mathematician I need proper explanation for trigonometric function extending beyond acute angle.
    (14 votes)
    Default Khan Academy avatar avatar for user
    • leaf red style avatar for user Noble Mushtak
      [cos(θ)]^2+[sin(θ)]^2=1 where θ has the same definition of 0 above.
      This is similar to the equation x^2+y^2=1, which is the graph of a circle with a radius of 1 centered around the origin. This is how the unit circle is graphed, which you seem to understand well.
      Based on this definition, people have found the THEORETICAL value of trigonometric ratios for obtuse, straight, and reflex angles. This value of the trigonometric ratios for these angles no longer represent a ratio, but rather a value that fits a pattern for the actual ratios. I hope this helped!
      Proof of [cos(θ)]^2+[sin(θ)]^2=1:
      https://www.khanacademy.org/cs/cos2sin21/6138467016769536
      (6 votes)
  • duskpin ultimate style avatar for user Scarecrow786
    At , shouldn't the point on the circle be (x,y) and not (a,b)? [Since horizontal goes across 'x' units and vertical goes up 'y' units--- A full explanation will be greatly appreciated]
    (6 votes)
    Default Khan Academy avatar avatar for user
    • aqualine ultimate style avatar for user Kyler Kathan
      It would be x and y, but he uses the letters a and b in the example because a and b are the letters we use in the Pythagorean Theorem a²+b² = c² and they're the letters we commonly use for the sides of triangles in general. It doesn't matter which letters you use so long as the equation of the circle is still in the form a²+b² = 1.
      (17 votes)
  • duskpin ultimate style avatar for user Mari
    This seems extremely complex to be the very first lesson for the Trigonometry unit. He keeps using terms that have never been defined prior to this, if you're progressing linearly through the math lessons, and doesn't take the time to even briefly define the terms. No question, just feedback.
    (8 votes)
    Default Khan Academy avatar avatar for user
  • duskpin ultimate style avatar for user Katie Huttens
    What's the standard position?
    (5 votes)
    Default Khan Academy avatar avatar for user
    • aqualine tree style avatar for user Ted Fischer
      A "standard position angle" is measured beginning at the positive x-axis (to the right). A positive angle is measured counter-clockwise from that and a negative angle is measured clockwise. (It may be helpful to think of it as a "rotation" rather than an "angle".)
      (7 votes)
  • leafers ultimate style avatar for user Rohith Suresh
    does pi sometimes equal 180 degree
    i saw it in a jee paper
    (3 votes)
    Default Khan Academy avatar avatar for user
  • marcimus pink style avatar for user contact.melissa.123
    why is it called the unit circle?
    (1 vote)
    Default Khan Academy avatar avatar for user
    • blobby green style avatar for user joe miller
      it's like I said above in the first post. If u understand the answer to this the whole unit circle becomes really easy no more memorizing at all!! It the most important question about the whole topic to understand at all! see my previous answer to Vamsavardan Vemuru
      (1 vote)

Video transcript

What I have attempted to draw here is a unit circle. And the fact I'm calling it a unit circle means it has a radius of 1. 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 1. So what would this coordinate be right over there, right where it intersects along the x-axis? Well, x would be 1, y would be 0. What would this coordinate be up here? Well, we've gone 1 above the origin, but we haven't moved to the left or the right. So our x value is 0. Our y value is 1. What about back here? Well, here our x value is -1. We've moved 1 to the left. And we haven't moved up or down, so our y value is 0. And what about down here? Well, we've gone a unit down, or 1 below the origin. But we haven't moved in the xy 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 the way I'm going to draw this angle-- I'm going to define a convention for positive angles. I'm going to say a positive angle-- well, the initial side of the angle we're always going to do along the positive x-axis. So you can kind of view it as the starting side, the initial side of an angle. And then to draw a positive angle, the terminal side, we're going to move in a counterclockwise direction. So positive angle means we're going counterclockwise. And 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 direction. So let me draw a positive angle. So a positive angle might look something like this. This is the initial side. And then from that, I go in a counterclockwise direction 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 where it intersects is b. And 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 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 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? You could view this as the opposite side to the angle. Well, this height is the exact same thing as the y-coordinate of this point of intersection. So this height right over here is going to be equal to b. 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 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 a. Now that we have set that up, what is the cosine-- let me use the 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 soh cah toa definition. That's the only one we have now. We are actually in the process of extending it-- soh cah toa definition of trig functions. And the cah 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. 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 soh part of our soh cah toa definition. It tells us that sine is opposite over hypotenuse. Well, the opposite side here has length b. And the hypotenuse has length 1. So 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. And b is the same thing as sine of theta. Well, that's interesting. We just used our soh cah toa definition. Now, can we in some way use this to extend soh cah toa? Because soh cah 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 make it part of a right triangle. But soh cah toa starts to break down as our angle is either 0 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 in it. It starts to break down. Let me make this clear. So sure, this is a right triangle, so the angle is pretty large. I can make the angle even larger and still have a right triangle. Even larger-- but I can never get quite to 90 degrees. At 90 degrees, it's not clear that I have a right triangle any more. It all seems to break down. And especially the case, what happens when I go beyond 90 degrees. So let's see if we can use what we said up here. Let's set up a new definition of our trig functions which is really an extension of soh cah toa and is consistent with soh cah toa. Instead of defining cosine as if I have a right triangle, and saying, OK, 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 is equal to the x-coordinate where we intersect, where the terminal side of our angle intersects the unit circle. And why don't we define 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 sine of theta. And so what would be a reasonable definition for tangent of theta? Well, tangent of theta-- even with soh cah toa-- 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.