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Sal draws the graphs of the sine and the cosine functions and analyzes their intersection points. Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
We're asked at how many points do the graph of y equals sine of theta and y equal cosine of theta intersect for theta between 0 and 2 pi. And 0 is less than or equal to theta which is less than or equal to 2 pi. So we're going to include 0 and 2 pi in the possible values for theta. So to do this I've set up a little chart for theta, cosine theta, and sine theta. And we can use this and the unit circle to hopefully quickly graph what the graphs of y equals sine theta and y equals cosine theta are. And then we could think about how many times they intersect and maybe where they actually intersect. So let's get started. So first of all, just to be clear, this is the unit circle. This is the x-axis. This is the y-axis. Over here we're going to graph these two graphs. So this is going to be the y-axis. And it's going to be a function of theta, not x, on the horizontal axis. So first let's think about what happens when theta is equal to 0. So when theta is equal to 0, you're at this point right over here. Let me do it in a different color. You're at this point right over here on the unit circle. And what to coordinate is that? Well that's the point 1 comma 0. And so based on that, what is cosine of theta when theta is equal to 0? Well cosine of theta is 1, and sine of theta is going to be 0. This is the x-coordinate at the point of intersection with the unit circle. This is the y-coordinate. Let's keep going. What about pi over 2? So pi over 2. We are right over here. What is that coordinate? Well that's now x is 0, y is 1. So based on that, cosine of theta is 0. And what is sine of theta? Well that's going to be 1. It's the y-coordinate right over. Now let's go all the way to pi. We're at this point in the unit circle. What is the coordinate? Well this is negative 1, 0. So what is cosine of theta? What's the x-coordinate here? Which is negative 1. And sine of theta is going to be the y-coordinate, which is 0. Now let's keep going. Now we're down here at 3 pi over 2. If we go all the way around to 3 pi over 2, what is this coordinate? Well this is 0, negative 1. Cosine of theta is the x-coordinate here. So cosine of theta is going to be 0. And what is sine of theta going to be? Well it's going to be negative 1. And then finally we go back to 2 pi, which is making a full revolution around the circle. We went all the way around and we're back to this point right over here. So the coordinate is the exact same thing as when the angle equals 0 radians. And so what is cosine of theta? Well that's 1. And sine of theta is 0. And from this we can make a rough sketch of the graph and think about where they might intersect. So first let's do cosine of theta. When theta is 0-- and let me mark this off. So this is going to be when y is equal to 1. And this is when y is equal to negative 1. So y equals cosine of theta. Let's see... theta equals 0. Cosine of theta equals 1. So cosine of theta is equal to 1. When theta is equal to pi 2, cosine of theta is 0. When theta is equal to pi, cosine of theta is negative 1. When theta is equal to 3 pi over 2, cosine of theta is equal to 0. That's this right over here. And then finally when theta is 2 pi, cosine of theta is 1 again. And the curve will look something like this. My best attempt to draw it. Make it a nice smooth curve. So it's going to look something like this. The look of these curves should look somewhat familiar at this point. So this is the graph of y is equal to cosine of theta. Now let's do the same thing for sine theta. When theta is equal to 0, sine theta is 0. When theta is pi over 2, sine of theta is 1. When theta is equal to pi, sine of theta is 0. When theta is equal to 3 pi over 2, sine of theta is negative 1. When theta is equal to 2 pi, sine of theta is equal to 0. And so the graph of sine of theta is going to look something like this. My best attempt at drawing it. So just visually, we can think about the question. At how many points do the graphs of y equals sine of theta and y equals cosine of theta intersect for this range for theta? For theta being between 0 and 2 pi, including those two points. Well, you just look at this graph. You see there's two points of intersection. This point right over here and this point right over here. Just between 0 and 2 pi. These are cyclical graphs. If we kept going, they would keep intersecting with each other. But just over this 2 pi range for theta, you get two points of intersection. Now let's think about what they are, because they look to be pretty close between 0 and pi over 2. And right between pi and 3 pi over 2. So let's look at our unit circle if we can figure out what those values are. It looks like this is at pi over 4. So let's verify that. So let's think about what these values are at pi over 4. So pi over 4 is that angle, or that's the terminal side of it. So this is pi over 4. Pi over 4 is the exact same thing as a 45 degree angle. So let's do pi over 4 right over here. So we have to figure out what this point is what. What the coordinates are. So let's make this a right triangle. And so what do we know about this right triangle? And I'm going to draw it right over here, to make it a little clear. This is a typical type of right triangle. So it's good to get some familiarity with it. So let me draw my best attempt. Alright. So we know it's a right triangle. We know that this is 45 degrees. What is the length of the hypotenuse? Well this is a unit circle. It has radius 1. So the length of the hypotenuse here is 1. And what do we know about this angle right over here? Well, we know that it too must be 45 degrees, because all of these angles have to add up to 180. And since these two angles are the same, we know that these two sides are going to be the same. And then we could use the Pythagorean Theorem to think about the length of those sides. So using the Pythagorean Theorem, knowing that these two sides are equal, what do we get for the length of those sides? Well, if this has length a, well then this also has length a. And we can use the Pythagorean Theorem. And we could say a squared plus a squared is equal to the hypotenuse squared. Is equal to 1. Or 2a squared is equal to 1a squared, is equal to 1/2. Take the principal root of both sides. a is equal to the square root of 1/2 which is the square root of 1, which is 1, over the square root of 2. We can rationalize the denominator here by multiplying by square root of 2 over square root of 2, which gives us a is equal to-- in the numerator-- square root of 2. And in the denominator, square root of 2 times square root of 2 is 2. So this length is the square root of 2. And this length is the same thing. So this length right over here is square root of 2 over 2. And this height right over here is also square root of 2 over 2. So based on that, what is this coordinate point? Well, it's square root of 2 over 2 to the right in the positive direction. So x is equal to square root of 2 over 2. And y is square root of 2 over 2 in the upwards direction, the vertical direction, the positive vertical direction. So it's also square root of 2 over 2. Cosine of theta is just the x-coordinate. So it's square root of 2 over 2. Sine of theta is just the y-coordinate. So you see immediately that they are indeed equal at that point. So at this point they're both equal to square root of 2 over 2. Now what about this point right over here, which looks right in between pi and 3 pi over 2. So that's going to be-- so this is pi, this is 3 pi over 2. It is right over here. So it's another pi over 4 plus pi. So pi plus pi over 4 is the same thing as 4 pi over 4 plus pi over 4. So this is the angle 5 pi over 4. So that's what we're trying to figure out. What are the value of these functions at theta equals 5 pi over 4? Well, there's multiple ways to think about it. You could even use a little bit of geometry to say, well if this is a 45 degree angle, then this right over here is also a 45 degree angle. You could say that the reference angle in terms of degrees is 45 degrees. And we could do a very similar thing. We can draw a right triangle. We know the hypotenuse is 1. We know that if this is a right angle, this is 45 degrees. If that's 45 degrees, then this is also 45 degrees. And we have a triangle that's very similar. They're actually congruent triangles. So hypotenuse is 1, 45, 90. We then know that the length of this side is square root of 2 over 2. And the length of this side is square root of 2 over 2. The exact same logic we used over here. So based on that, what is the coordinate of that point? Well, let's think about the x value. It's square root of 2 over 2 in the negative direction. We have to go square root of 2 over 2 to the left of the origin. So it's negative square root of 2 over 2. This point on the x-axis is negative square root of 2 over 2. What about the y value? Well, we have to go square root of 2 over 2 down, in the downward direction, from the origin. So it's also negative square root of 2 over 2. So cosine of theta is negative square root of 2 over 2. And sine of theta is also negative square root of 2 over 2. And so we see that we do indeed have the same value for cosine of theta and sine of theta right there. They are both equal to the negative square root of 2 over 2.