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# Proof of the law of sines

CCSS.Math:

## Video transcript

I will now do a proof of the law of sines so let's say let me draw an arbitrary triangle let's see that's one side right there and then I get another side here and I'll try to make it look a little strange so you realize that can apply to any triangle and let's say we know we know the following information we know this angle well actually I'm not going to say what we know or don't know but I want to the law of sines is just a relationship between different angles and different sides so let's say that this angle right here is alpha the side here is a the length here is a let's say that this side here is beta and that the length here is B all right B beta is just you know B with a long end there so let's see if we can find a relationship that connects a and B and alpha and beta so what can we do so let me draw and hopefully that that relationship we find will be the law of sines otherwise I have I would have to rename this video so let's let me draw an altitude here so I think that's the proper term so if I if I just draw a line from this side coming straight down and it's going to be perpendicular to this bottom side which I haven't labeled to put you know probably if I have to label it probably label at C because that's a and B and this is going to be a 90-degree angle this is going to be a 90-degree angle and I don't know I don't know the length of that I don't know anything about it all I know is I I took a look I I went from this vertex and I dropped a line that forms a 90 that's perpendicular to this other side so what can we do with this line well let me just say that it has length x the length of this line is X so we can can we find a relationship between a the length of this line X and beta well sure let's see let me find an appropriate appropriate color okay that's I think a good color so what's a relationship for if we look at this angle right here beta X is opposite to it and a is the hypotenuse if we look at this right try right here right so what deals with opposite and hypotenuse so you're always whenever we do trigonometry we should always just write sohcahtoa at the top of the page sohcahtoa so what deals with opposite and hypotenuse sine right so and you should probably guess that because I'm proving the law of sines so let's see the sine of beta the sine of beta is equal to the opposite over the hypotenuse so it's equal to this opposite which is x over the hypotenuse which is a in this case and if we wanted to solve for X and I'll just do that because it'll be convenient later we can multiply both sides of this equation by a and you get a sine of beta is equal to X fair enough that got us someplace well let's see if we can find a relationship between alpha B and X well similarly if we look at this right triangle because this is also a right triangle of course what deals X here relative to alpha is also the opposite side and B now is the hypotenuse so we can also write we can also write that sine of alpha let me do it in a different color sine of alpha is equal to sine of alpha is equal to opposite over hypotenuse opposite over hypotenuse the opposite is X and the hypotenuse is B the hypotenuse is B and let's solve for X again just to do it multiply both sides by B and you get B sine of alpha is equal to X so now what do we have we have two different ways that we solved for this this thing that I drop down from this side this X right we have a sine of beta is equal to X and that B sine of alpha is equal to X well if they're both equal to X and they're both equal to each other so let me write that down let me write that down in a soothing color so we know that a sine of beta is equal to write a sine of beta is equal to X which is also equal to B sine of theta I'm sorry B sine of alpha B sign of alpha if we divide both sides of this equation by a what do we get we get sine of beta right because the a on this side cancels out is equal to B sine of alpha over a and if we divide both sides of this equation by B we get sine of beta over B is equal to sine of beta over is equal to sine of alpha over a so with that this is the law of sines the ratio between the sine of beta and its opposite side and it's you know the side that it corresponds to this B is equal to the ratio of the sine of alpha and its opposite sign and then you know a lot of times in the books if let's say if this angle was theta and this was C then they you know they would also write that's also equal to the sine of theta sine of theta over C and and the proof of adding this here is identical I mean we picked B arbitrarily B is the side we could have done the exact same thing with theta and C but instead of drawing the alt dropping the altitude here we would have to drop one of the other altitudes and you I think you could figure out that part but the important thing is we have this ratio and of course you could have written it you since it's a ratio you could flip both sides of the ratio you could write it B over the sine of B is equal to a over the sine of alpha and this is useful because if you know if if you know one side and it's corresponding angle the angle opposite it that kind of opens up into that side and say you know the other side then you could figure out the angle that opens up into it or I mean well you could fit if you know three of these things you can figure out the fourth and that's what's useful about about the law of sines so maybe now I will do a few law of sines word problems I'll see you in the next video