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Sine & cosine identities: periodicity

Sal finds trigonometric identities for sine and cosine by considering angle rotations on the unit circle. Created by Sal Khan.

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  • purple pi purple style avatar for user Sam Hyden
    What are some other relationships besides the one that Sal came up with at the end of the video?
    (32 votes)
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    • leaf red style avatar for user Noble Mushtak
      Here are some trigonometric relationships that can be found by playing around with the unit circle:
      sin(pi-θ)=sin(θ)
      cos(pi-θ)= -cos(θ)
      sin(θ+pi)= -sin(θ)
      cos(θ+pi)= -cos(θ)
      cos(θ+pi/2)= -sin(θ)
      sin(-θ)= -sin(θ)
      cos(-θ)=cos(θ)

      I hope these relationships help you in trig!
      (187 votes)
  • blobby green style avatar for user Brandon Hoffman
    Is sin always related to the y axis? I'm confused because if we look at angle theta + pi/2, why is it that sin doesn't have the opposite/hypotenuse definition? If sin is opposite side/hypotenuse wouldn't the sin of theta + pi/2= sin of theta? The opposite side of the "triangles" is the same isn't it?
    (33 votes)
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  • male robot hal style avatar for user Brett Daniels
    Is it me, or did Sal mistake his notation in the final part of the video in concluding that cos(theta) = sin(theta+pi/2)? Because that is not a right triangle, and would need to be analyzing its complimentary angle (90-theta) since 90-theta+theta+pi/2=180 degrees. Since sin(90-theta) = cos(theta), and cos(90-theta) = sin(theta) this seems to be the right way... anyone else notice this or am I mistaken?
    (12 votes)
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    • piceratops ultimate style avatar for user Just Keith
      Sine, cosine and the other functions are not just defined for right angles, though the simple definitions you start with for these functions only work for the acute angles of right triangles.

      But, yes, cos x = sin(x + ½π)
      It is also true that cos x = sin(½π - x)
      Thus, it is true that
      sin(½π - x) = sin(x + ½π)

      It is also true that cos(½π-x) = sin(x)
      and that cos(x- ½π) = sin(x)
      So, cos(½π-x) = cos(x- ½π)

      Sine and cosine are both periodic functions that are identical except for being shifted ½π radians out of phase. Thus, there are a number of ways you can shift them around to be in phase and therefore equal.

      Note: Do not mix degrees and radians. Once you can use radians, you should just drop degrees altogether because in math that isn't too far down the road, degrees just won't work, you have to use radians.
      (33 votes)
  • male robot hal style avatar for user Marioland
    At Sal says that the length of the magenta line is Sine theta plus Pi. How did he come to that?

    Wouldn't the length of that line be equal to Cos theta since we are rotating the triangle?
    (24 votes)
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  • duskpin ultimate style avatar for user Laurent_Fong
    I think there is a good way for us to remember these trig relationships by a "Chinese Magical term 奇变偶不变,符号看象限”, which means that:
    1. the change between sin and cos is based on the angle (x + θ) (in this case, if the number "x" is the 90 degree's odd multiple, such as 270 degree that is 3 times of 90 degree, the sin will be changed into cos while the cos will be changed into sin.
    For example: sin(θ) = cos(270 + θ) because "270 = 90 x 3, 3 is odd"
    cos(θ) = sin (450 + θ) because " 450 = 90 x 5, 5 is odd"


    2. On the other hand, if the y, in the angle (y + θ), is the even multiple to 90 degree, such as pi, 180 degree, 360 degree, etc., the sin or cos will not be changed.
    For example: sin(θ) = sin (180 + θ), because "180 = 90 x 2, 2 is even"
    cos(θ) = cos (540 + θ), because "540 = 90 x 6, 6 is even"


    3. the positive or negative is based on the initial degree's sign, such as angle (180 + θ), in this case, if θ is an angle in the first quadrant, which means that both of sin(θ) and cos(θ) are positive, the sin(180 + θ) will be sin(θ) without being changed, because " θ is in the first quadrant and 180 = 2 x 90, 2 is even, according to the previous sentence.
    For example:
    sin120 = sin (90 + 30) = cos (90 x 1 + 30), because 1 is odd, and sin120 is in the second quadrant, corresponding to positive sign.
    sin600 = sin(6 x 90 + 60) = sin60, because 6 is even, and sin60 is in the first quadrant, corresponding to the positive sign.
    -cos150 = -cos(90 x 1 + 60) = -sin60, because 1 is odd, and -cos150 is in the third quadrant, corresponding to the negative sign.



    At the end, thanks for the Khan Academy's service about SAT and other tremendously meaningful programs!
    (18 votes)
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  • hopper cool style avatar for user Big Brain Yuvraj 🧠
    Why do people name angles with Greek letters?
    (5 votes)
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  • piceratops ultimate style avatar for user Richard Sanchez
    does cos(θ+π/2) = sin (θ) ?
    (4 votes)
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    • piceratops ultimate style avatar for user Fai
      Not quite. Remember the direction in which you want the function to move. Since you want to move the cosine function forward, you have to subtract the pi/2 from the theta value.

      So, sin(theta) = cos(theta - pi/2).

      What you have there will be -sin(theta)
      (14 votes)
  • blobby green style avatar for user birdlion88
    At there's a dotted horizontal line above the second quadrant which would have supposedly made a triangle. How come sin wasn't on that dotted line and instead was on the y-axis? Shouldn't it be on that dotted line?
    (6 votes)
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    • aqualine ultimate style avatar for user Rafeea Rahman
      That triangle is not related to the sin that we are looking for. We are looking for the sin(θ+π/2), and we can find it by looking at the y-coordinate of the point on the ray which intersects the circle. This y-coordinate is also the vertical distance from said point to the origin. This is exactly the line that was highlighted in pink.

      The horizontal line you mentioned is actually sin (θ). Imagine that triangle you pointed out being rotated clockwise by 90° at the origin. The two triangles map onto one another.
      (5 votes)
  • starky sapling style avatar for user Fomentia
    Sal says that the unit circle definition is an extension of soh-cah-toa, but wouldn't it be the other way around? Isn't soh-cah-toa an extension of the unit circle definition?
    (1 vote)
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    • aqualine ultimate style avatar for user AquaGon
      I guess you could think of it both ways. Both definitions are derivable from the other definition. For example, you can say:
      Sin(x) = Opposite/Hypotenuse
      (In a unit circle, Hypotenuse=1, so)

      > Sin(x) = Opposite/1
      > Sin(x) = Opposite
      In this case, the length of the side opposite the central angle = the y length of the triangle.
      >> that means that the y coordinate (Where the Hypotenuse intersects the circle) = Sin(x)
      >>>the same logic applies for cos(x)

      The reason some people say that the unit circle definition is an extension of Soh-Cah-Toa is mostly because Soh-Cah-Toa (I believe) was defined first, and the Unit circle definition is applicable to more scenarios(Any triangle versus only right triangles). The Unit circle definition EXPANDS upon the Soh-Cah-Toa definition. It takes you more places.
      (11 votes)
  • duskpin ultimate style avatar for user Sajiya70
    In the video, Sal proved that:

    sin(π/2 + θ) = cos θ

    But, in my textbook, the identity given goes like this;

    sin(π/2 - θ) = cos θ

    So does that mean that sin(π/2 - θ) = sin(π/2 + θ) or is there a mistake somewhere?
    (5 votes)
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Video transcript

Let's say that I've got some angle theta, some angle theta right over here. And I'm drawing it on our unit circle with a typical convention that we started with a ray that's along the positive X axis, and the terminal side of this angle, is the terminal side of the angle, where it intersects the unit circle, determines essentially the sine and cosine of that theta, so the cosine of theta is the X... is the... let me just set a color I haven't used before. The cosine of theta is the X coordinate of where this terminal ray intersects the unit circle. Or another way of thinking about it is the cosine of theta is the length of what I'm drawing in purple right over here. It's this length. That length right over there is cosine of theta, and the sine of theta is the Y coordinate. Or another way of thinking about it the sine of theta is the length of this line right over here. The how high you are above the X axis, that is essentially the Y coordinate, and so the length of that is sine theta. And this makes sense, this actually shows why the unit circle definition is an extension of the Soh Cah Toa definition. Remember, Soh Cah Toa. Let me write it down. Soh Cah Toa. Soh Cah Toa. So sine is opposite over hypotenuse. So if I want to do the sine of theta, what's it going to be? So if I think about the sine of theta, sine of theta by the Soh Cah Toa definition, it's going to be equal to the length of the opposite side. Well, we're saying that that's sine of theta, it's sine of theta, over the hypotenuse. Well the hypotenuse here, this is a unit circle, so it's going to be one. So this shows that this is consistent. Or another way of thinking about it, sine of theta is equal to the opposite side over the hypotenuse. In this case it's going to be equal to the opposite side, and what's the hypotenuse? This is a unit circle, so it's going to be one. In this case, sine of theta is equal to the length of the opposite side. The length of the opposite side is equal to sine theta. And same exact logic. The cosine of theta is equal to adjacent over hypotenuse, is equal to adjacent over hypotenuse. And so that's... since the hypotenuse is equal to one, it's just the length of the adjacent side, so cosine of theta is the length of the adjacent side. So this is all a little bit of review, just showing how the unit circle definition is an extension of the Soh Cah Toa definition. But now let's do something interesting. This is the angle theta. Let's think about the angle theta plus pi over two. So the angle theta plus pi over two. So if I were to essentially add pi over two to this, I'm going to get a ray that is perpendicular to the first ray, pi over two. If we think in degrees, pi over two radians, so when I say theta plus pi over two, I'm talking in radians. Pi over two radians is equivalent to 90 degrees. So we're essentially adding 90 degrees to it. So this angle right over here, that angle right over here is theta plus pi over two. Now, what I want to explore in this video, and I guess this is the interesting part of the video, is can we relate sin of theta plus pi over two to somehow sine of theta or cosine of theta? I encourage you to pause this video and try to think this through on your own before I work it out. Well let's think about what sine of theta plus pi over two is. We know from the unit circle definition, the sine of this angle, which is theta plus pi over two, is the Y coordinate. It's that, it's this value right over here. Or another way of thinking about it, it's the length of this line in magenta. This right over here is the sine of theta, plus pi over two. So that right over there. Now how does that relate to what we have over here? Well when you look at it, it looks like we just took this triangle, and we just kind of... we rotated it. We rotated it counter clockwise by 90 degrees, which essentially what we did do. Because we took this terminal side, and we added 90 degrees to it, or pi over two radians. And if you want to get a little bit more rigorous about it, if this whole white angle here is theta plus pi over two, and the part that's in the first quadrant is pi over two, then this part right over here, that must be equal to theta. And if we think about it, if we try to relate the side this side that I've put in magenta relative to this angle theta using the Soh Cah Toa definition, here, relative to this angle theta in yellow, this is the adjacent side. So let's think about it a little bit. So if we were.. so what deals with the adjacent and the hypotenuse, and in this case of course our hypotenuse has length one, this is a unit circle, what cosine deals with adjacent and hypotenuse? So we could say that the cosine of this theta so the cosine of that theta is equal to the adjacent side, the length of the adjacent side which we already know is sine of theta plus pi over two. Let me write it this way. Sine of theta plus pi over two. over the hypotenuse. Over the hypotenuse, which is just one, so that doesn't change its value. So that was pretty neat. Just like that, we were able to come up with a pretty neat relationship between cosine and sine. The cosine of theta is equal to sine of theta plus pi over two, or you could say sine of theta plus pi over two is equal to cosine of theta. Now what I encourage you to do is, after this video, see if you can come up with other results. Think about what happens to... what sine of theta relates to? Or what cosine of theta plus pi over two might relate to? So I encourage you to explore that on your own.