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Sine and cosine come from circles

Sine and cosine can be generated by projecting the tip of a vector onto the y-axis and x-axis as the vector rotates about the origin.  Created by Willy McAllister.

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  • starky sapling style avatar for user Angela Zou
    Will drawing a circle with a bigger radius allow us to find angles that have sine or cosine values that are greater than 1?
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      Clever idea, but no. Trig functions (sine, cosine, tangent) are defined as ratios. Sine is "opposite over hypotenuse" (the SOH of SOHCAHTOA). When we draw the triangle inside a unit circle the hypotenuse is automatically 1 at any angle. That means the sine of an angle is simply the length of the "opposite" leg of the triangle (opposite / 1). If you make the circle radius = 2 it makes both O and H twice as long, but the ratio stays the same. Nice try. I like the way you think.
      (4 votes)
  • blobby green style avatar for user Annette Mauracher
    Hello, we are starting the curves of sine and cosine at school, is this what I need to know for an understanding of it, I mean is this the basics for starting the sine and cosine graphs/curves? Thank you
    (1 vote)
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    • spunky sam orange style avatar for user Willy McAllister
      Short answer: Yes.

      When sine and cosine are first revealed in Trigonometry class they are taught as ratios of the the sides of right triangles. The memorization tool is "SOA CAH TOA".

      If you fit the right triangle inside a circle, you can slide the tip of the triangle around the circle and think about sin and cos of a point somewhere along the circle. It all means the same thing, just a slightly different mental picture.

      Using this circle perspective is really handy. In the next few videos we let the point on the circle move as time goes by, so you get this orbiting point. That turns out to be a super useful way to think about Signals. (sound, radio waves, light)

      Meanwhile, in Trig class you will use the right triangle viewpoint to prove all sorts of useful properties and identities of trig functions.
      (3 votes)
  • leaf orange style avatar for user Ali Reda
    Can you please explain where did the sine function come from? From the unit circle? or exponentially?
    (0 votes)
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  • duskpin ultimate style avatar for user Mahrukh
    Why does the sine and the cosine waves look like waves and not like semicircles joining each other in succession if they are related to circles?
    (1 vote)
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Video transcript

- [Voiceover] Now I'm gonna clear off the screen here and we're gonna talk about the shape of the sine function. Let's do that. This is a plot of the sine function where the angle, theta, this is the theta axis in this plot, where theta has been plotted out on a straight line instead of wrapped around this circle. So if we draw a line on here, let's make this circle a radius one. So if I draw this line up here, it's on a unit circle, the definition of sine of theta, this will be theta here, is opposite over hypotenuse. So this is the opposite side and that distance is the opposite leg of that triangle, is this value right here. So sine of theta is actually equal to y over the hypotenuse and the hypotenuse is one in all cases around this. So, if I plot this on a curve, this is an angle and I basically go over here and plot it like that. And then as theta swings around the circle, I'm gonna plot the different values of y. If it comes over this way, down here like this, right, you can see that, that plots over there like that. Now when the angle gets back all the way to zero, of course, the sine function comes all the way back to zero and then it repeats again as our vector swings around the other way. So the sine of two pi is zero, just like the sine of zero. So every two pi, if I go off the screen, every two pi comes back and repeats to zero. So now I wanna do the same thing with the cosine function that we did with sine, where we project the projection of this value onto this time the cosine curve down here. This has the cosine curve with time going down on the page. And our definition of cosine was adjacent over hypotenuse. Hypotenuse is one in our drawing. So cosine of theta equals adjacent which is x, the x value, divided by hypotenuse which is one. So in this diagram, the cosine of theta is actually the x value which is this x right here. So let me clean this off for a second. And we'll start at the beginning. Let's start with the radius pointing straight sideways. And we know that cosine of theta equals zero is one. So if I drop that down, if I project that down onto the angle zero, that's this point right here on the curve. Now as we roll forward, we go to a higher angle, this projection now moves to here on the curve. When the arrow is straight up, we are at this point right here, we go back to the axis. If we go continue on, this projects down here. We're moving this radius vector around in a circle like this. And actually this one will be at the same point as before, as the one above, but it'll be on this part of the curve here. And when we get back to zero again, the projection is to this point here. So that's a way to visualize the cosine curve getting generated by a vector rotating around this circle. The cosine comes out the bottom because it's the projection on the x-axis, and when we did the sine, it was the projection on the y-axis, produced the sine wave when we went this way. So I like to visualize this because this rotating vector is a really simple and powerful idea, and we can see how it actually generates, it's a way to generate sine and cosine waves. And you can see how sort of naturally they come out at different phases, right. The sine starts at zero and the cosine starts at one. With this way of drawing it, you could see why that happens. So this relationship between circles and rotating vectors and sines and cosines is a very powerful idea. We're really gonna take advantage of this.