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Solving cos(θ)=1 and cos(θ)=-1

Sal solves the equations cos(θ)=1 and cos(θ)=-1 using the graph of y=cos(θ). Created by Sal Khan and Monterey Institute for Technology and Education.

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  • marcimus pink style avatar for user devanshi
    What is exactly happening at and why is 'y' taken as equal to cosQ?
    (28 votes)
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    • leaf blue style avatar for user Brandon Kennedy
      im learning this myself and this is what i have so far.

      "y" is taken as the equal to cos(theta) because that is what you are measuring. remember the "x" axis is actually the "theta" axis. you you are graphing where cos(theta) is when theta is at a specified value. so x-axis=theta and y-axis=cos(theta). if you were trying to find the sin(theta), this would be a different graph, then that would make y-axis=sin(theta).

      i was at first confused from the fact that the coordinates were supposed to be (cos, sin), but then realized that this only applied to the unit circle.

      at he is showing the direction you should travel on the unit circle to get the correct coordinates for the graph. if you move the wrong direction, you aren't getting the correct numbers. you can really tell when you start graphing.
      (46 votes)
  • orange juice squid orange style avatar for user FishHead
    Is there some sort of intuition as to why cos is the x of the unit circle and sin is the y?
    (15 votes)
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    • male robot donald style avatar for user Jeremy
      FishHead,

      Well, yes, it's all based on how we define the angle that we are interested in when we are using the unit circle (most people call that angle θ, theta). Just as a convention, we define θ as the angle from the X axis, to the line that goes from the origin to the point on the circle we are interested in. So when we talk about the point on the unit circle at 45 degrees, ( or pi/4 radians), that's 45 degrees from the positive X axis.

      If we have some point on the unit circle, and we draw a line from the origin to that point, that gives us 1 leg of a triangle. If we use the X axis as the second leg of the triangle, and then draw a line that goes through the point on the circle to the x axis (and we make sure the line is perpendicular to the x axis), that gives us the third leg of our right triangle. If you do this, you'll see that the angle, θ, that I was talking about before is one of the three angles of this right triangle. It is the angle at the point of the triangle that touches the origin. So if we take the cosine of that angle, that is equal to the adjacent side of that triangle divided by the hypotenuse. But notice that the adjacent side of the triangle is just the x coordinate for the original point on the unit circle that we are interested in, and the hypotenuse of the circle is just 1. So the adjacent side over the hypotenuse is just the x coordinate of the point on the circle divided by 1.

      Similarly, the sign of the angle θ is the opposite side of the triangle that we drew, divided by the hypotenuse. But this is just the y coordinate of the point on the circle, divided by 1… which is just the y-coordinate of the point on the circle.

      If this doesn't make sense, it may help to draw a pair of x and y axes, and draw a circle on the paper centered on the origin with a radius of 1. Then, pick a point on the circle (and I would pick a point in quadrant 1 until you get comfortable with the concepts), and draw the appropriate right triangle.
      (31 votes)
  • male robot hal style avatar for user Quiqué
    I wonder does cosθ² + sinθ² = 1?
    (4 votes)
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    • piceratops ultimate style avatar for user Just Keith
      If you mean:
      cos (θ²) + sin(θ²), then that is NOT equal to 1,
      except for a few special angles such as θ=√(2π), θ=0 or θ= ½√(2π)

      If you mean: (cos θ )² + (sin θ)² = 1
      Which is usually written as: cos² (θ) + sin²( θ) = 1
      Then that is true.
      (26 votes)
  • leafers seed style avatar for user Helen
    At about it states that when cos= 1 then theta is 0; I'm confused, I thought cos equalled one at 2π.
    (3 votes)
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  • orange juice squid orange style avatar for user Cindy
    What is Theta?
    (8 votes)
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  • spunky sam blue style avatar for user Eesh Starryn
    sal can u make another video on this? The problems in the exercises are harder than the videos
    (7 votes)
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  • aqualine ultimate style avatar for user Ainul Jeffery
    Why don't we have any lessons on Secant, Cosecant and Cotangent?
    I have no visual idea of what they are.
    (3 votes)
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  • blobby green style avatar for user ibrahimrible1
    Why does the sine and cosine graph range between 1 and -1
    (2 votes)
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  • male robot johnny style avatar for user Mohamed Ibrahim
    In the exercise after this video, when the question asked to choose all the values for the equation sin (0.65)= x, the answer told to find the values between x is greater than -π and less than or equal to π. Why didn't it say greater than or equal to -π ?
    (2 votes)
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  • blobby green style avatar for user DODO KIM
    Is cos 180 = -1 or 1

    In my math text book it says 1 but the internet says -1
    (1 vote)
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Video transcript

- [Instructor] In the graph below, for what values of theta does cosine of theta equal one, and for what values of theta does cosine of theta equal negative one? And they've very nicely graphed it for us, the horizontal axis is the theta axis, and the vertical axis is the y-axis, and so this is the graph of y is equal to cosine of theta. And it makes sense with a unit circle definition, and I'll just make sure that we're comfortable with that, because with our unit circle definition, so let me draw ourselves a unit circle, and I'm just going to draw it very roughly, just so that we get the general idea of what's going on here. When theta is equal to zero, we're at this point right over here on the unit circle. Well what's the x-coordinate of that point? Well it's one, and you see when theta's equal to zero, on this graph, cosine of theta is equal to one. When theta is equal to pi over two, we're at this point on the unit circle, and the x-coordinate is what? Well, the x-coordinate there is zero. And you see once again, when we're at pi over two, the x-coordinate is zero, so this is completely consistent with our unit circle definition. As we move in the rightward direction, we're moving counterclockwise around the unit circle, and as we move in the leftward direction, we're moving counter, sorry. If we move in the rightward direction, we're moving counterclockwise, and as we're moving in the leftward direction along the axis in the negative angles, we're moving in the clockwise, we're moving in the clockwise direction around our unit circle. So let's answer their question. For what values of theta does cosine of theta equal one? Well we can just read the graph right over here. It equals one, so cosine of theta equals one, cosine of theta equals one at, at theta is equal to, well, we see it right over here. Theta is equal to zero, theta is equal to, well we've gotta go all the way again to two pi, two pi, but then it just keeps going on and on, and it makes sense. Theta equaled, or sorry, cosine of theta, the x-coordinate on this unit circle equaled one right when we were at zero angle, and we had to go all the way around the circle to get back to that point, two pi radians. But then it'll be again when we get to four pi radians, and then six pi radians, so two pi, four pi, six pi, and I guess you could see the pattern here. We're gonna keep hitting cosine of theta equals one every two pi, so you could really kind of view this as every multiple of two pi. Two pi n, where n is an integer, n is integer... is an integer. And that applies also for negative values. If you're going the other way around, if we're going the other way around, we don't get back until we get to negative two pi. Notice we were at zero, and then the next time we're at one again is at negative two pi, and then negative four pi, and then over and over and over again. But this applies, if n is an integer, n can be a negative number, and so we get to all of the negative values of theta where cosine of theta is equal to one. Now let's think about when cosine of theta is equal to negative one. So cosine of theta is equal to negative one at theta is equal to, well we can just look at this graph right over here. Well when theta is equal to pi, when theta is equal to pi, and let's see, well, it kinda goes off this graph, but this graph would keep going like this, would keep going like this, and you'd see it would also be at three pi. And you can visualize it over here. Theta, cosine of theta is equal to negative one when we're at this point on the unit circle. So that happens when we get to pi radians, and then it won't happen again until we get to two pi, three pi radians, three pi radians. And it won't happen again until we go to two pi, until we add another two pi, until we make one entire revolution, so then that's going to be five, five pi radians. And you can keep going on and on and on, and that's also true in the negative direction, so if we take two pi away from this, so if we were here, and if we go all the way around back to negative pi, it should also be the case, and you actually see it right over here on the graph. So you could think about this as two pi, two pi n plus pi, or you could view it as two n plus one, or two n plus one times pi, where pi is, sorry, where n is an integer. Let me right that a little bit neater, n is integer. At every one of those points, cosine, or for every one of these thetas, cosine of theta is going to keep hitting negative one over and over again. And you see it, it goes it goes from one bottom, where you can kind of valley to the next valley, it takes two pi to get to the next valley, two pi to get to the next valley. And that was also the same thing for the peaks. It took two pi to go from the top of one hill to the top of the next, and then two pi again to the top of the hill after that.