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Proof of the law of cosines

Sal gives a simple proof of the Law of cosines. Created by Sal Khan.
Video transcript
In the last video, we had a word problem where we had-- we essentially had to figure out the sides of a triangle, but instead of, you know, just being able to do the Pythagorean theorem and because it was a right triangle, it was just kind of a normal triangle. It wasn't a right triangle. And we just kind of chugged through it using SOHCAHTOA and just our very simple trig functions, and we got the right answer. What I want to do now is to introduce you to something called the law of cosines, which we essentially proved in the last video, but I want to kind of prove it in a more-- you know, without the word problem getting in the way, and I want to show you, once you know the law of cosines, so you can then apply it to a problem, like we did in the past, and you'll do it faster. I have a bit of a mixed opinion about it because I'm not a big fan of memorizing things. You know, when you're 40 years old, you probably won't have the law of cosines still memorized, but if you have that ability to start with the trig functions and just move forward, then you'll always be set. And I'd be impressed if you're still doing trig at 40, but who knows? So let's go and let's see what this law of cosines is all about. So let's say that I know this angle theta. And let's called this side-- I don't know, a. No, let's call this side b. I'm being a little arbitrary here. Actually, let me stay in the colors of the sides. Let's call that b and let's call this c, and let's call this side a. So if this is a right triangle, then we could have used the Pythagorean theorem somehow, but now we can't. So what do we do? So we know a-- well, let's assume that we know b, we know c, we know theta, and then we want to solve for a. But, in general, as long as you know three of these, you can solve for the fourth once you know the law of cosines. So how can we do it? Well, we're going to do it the exact same way we did that last problem. We can drop a line here to make-- oh, my God, that's messy. I thought I was using the line tool. Edit, undo. So I can drop a line like that. So I have two right angles. And then once I have right triangles, then now I can start to use trig functions and the Pythagorean theorem, et cetera, et cetera. So, let's see, this is a right angle, this is a right angle. So what is this side here? Let me pick another color. I'm probably going to get too involved with all the colors, but it's for your improvement. So what is this side here? What is the length of that side, that purple side? Well, that purple side is just, you know, we use SOHCAHTOA. I was just going to write SOHCAHTOA up here. So this purple side is adjacent to theta, and then this blue or mauve side b is the hypotenuse of this right triangle. So we know that-- I'm just going to stick to one color because it'll take me forever if I keep switching colors. We know that cosine of theta-- let's call this side, let's call this kind of subside-- I don't know, let's call this d, side d. We know that cosine of theta is equal to d over b, right? And we know b. Or that d is equal to what? It equals b cosine theta. Now, let's call this side e right here. Well, what's e? Well, e is this whole c side-- c side, oh, that's interesting-- this whole c side minus this d side, right? So e is equal to c minus d. We just solved for d, so side e is equal to c minus b cosine of theta. So that's e. We got e out of the way. Well, what's this magenta side going to be? Well, let's call this magenta-- let's call it m from magenta. Well, m is opposite to theta. Now, we know it. We've solved for c as well, but we know b, and b is simple. So what relationship gives us m over b, or involves the opposite and the hypotenuse? Well, that's sine: opposite over hypotenuse. So we know that m over b is equal to sine of theta. We know that-- let me go over here. m over b, right, because this is the hypotenuse, is equal to sine of theta, or that m is equal to b sine of theta, right? So we figured out m, we figured out e, and now we want to figure out a. And this should jump out at you. We have two sides of a right triangle. We want to figure out the hypotenuse. We can use the Pythagorean theorem. The Pythagorean theorem tells us a squared is equal to m squared plus e squared, right? Just the square of the other two sides. Well, what's m squared plus e squared? Let me switch to another color just to be arbitrary. a squared is equal to m squared. m is b sine of theta. So it's b sine of theta squared plus e squared. Well, e we figure out is this. So it's plus c minus b cosine theta squared. Now, let's just chug through some algebra. So that equals b sine-- b squared sine squared of theta. Sine squared of theta just means sine of theta squared, right? Plus, and we just foiled this out, although I don't like using foil. I just multiply it out. c squared minus 2cb cosine theta plus b squared cosine theta, right? I just expanded this out by multiplying it out. And now let's see if we can do anything interesting. Well, if we take this term and this term, we get-- those two terms are b squared sine squared of theta plus b squared cosine-- this should be squared there, right, because we squared it. b squared cosine squared of theta, and then we have plus c squared minus 2bc cosine theta. Well, what does this simplify to? Well, this is the same thing as b squared times the sine squared theta plus cosine squared of theta. Something should be jumping out at you, and that's plus c squared minus 2bc cosine theta. Well, this thing, sine squared plus cosine squared of any angle is 1. That's one of the earlier identities. That's the Pythagorean identity right there. So this equals 1, so then we're left with-- going back to my original color. We're almost there-- a squared is equal to-- this term just becomes 1, so b squared. We're just left with a b squared plus c squared minus 2bc cosine of theta. That's pretty neat, and this is called the law of cosines. And it's useful because, you know, if you know an angle and two of the sides of any triangle, you can now solve for the other side. Or really, if you want to, if you know three sides of a triangle, you can now solve for any angle, so that also is very useful. The only reason why I'm a little bit, you know, here, there, is I don't-- if you are in trigonometry right now and you might have a test, you should memorize this because it'll make you faster, and you'll get the answer right quicker. I'm not a big fan of just memorizing it without knowing where it came from, because a year from now or two years from now when you go to college and it's been four years since you took trigonometry, you probably won't have this memorized. And if you face a trig problem all of a sudden, it's good to kind of get there from scratch. With that said, this is the law of cosines, and if you use the law of cosines, you could have done that problem we just did a lot faster because we just-- you know, you just have to set up the triangle and then just substitute into this, and you could have solved for a in that ship off-course problem. I'll see you in the next video.