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Sine & cosine identities: periodicity

Video transcript
Let's say that I've got sum angle theta, sum 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.