# Proof: Law of sines | Trig identities and examples | Trigonometry | Khan Academy

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I will now do a proof

of the law of sines. So, let’s see, let me draw

an arbitrary triangle. That’s one side right there. And then I’ve got

another side here. I’ll try to make it look a

little strange so you realize it can apply to any triangle. And let’s say 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

the law of sines is just a relationship between different

angles and different sides. Let’s say that this angle

right here is alpha. This 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. Beta is just 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? And hopefully that

relationship we find will be the law of sines. Otherwise, I would have

to rename this video. So let me draw an

altitude here. I think that’s the proper term. 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, but I’ll

probably, if I have to label it, probably label it C,

because that’s A and B. And this is going to

be a 90 degree angle. I don’t know the

length of that. I don’t know anything about it. All I know is I went from this

vertex and I dropped a line 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. 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 color. OK. That’s, I think, a good color. So what’s the relationship? 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 triangle right here, right? So what deals with

opposite and hypotenuse? Whenever we do trigonometry, we

should always just right soh cah toa at the top of the page. Soh cah toa. So what deals with

opposite of hypotenuse? Sine, right? Soh, and you should probably

guess that, because I’m proving the law of sines. So the sine of beta is

equal to the opposite over the hypotenuse. 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, x here, relative to alpha, is also the opposite

side, and B now is the hypotenuse. So we can also write that sine

of alpha — let me do it in a different color — is equal

to opposite over hypotenuse. The opposite is x and

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 thing that I dropped down from this

side, this x, right? We have A sine of

beta is equal to x. And then B sine of

alpha is equal to x. Well, if they’re both equal

to x, then 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 x, which is also equal to B sine of beta —

sorry, B sine 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 alpha over A. So this is the law of sines. The ratio between the sine of

beta and its opposite side — and it’s the side that it

corresponds to, this B — is equal to the ratio of the sine

of alpha and its opposite side. And a lot of times in the

books, let’s say, if this angle was theta, and this was C, then

they would also write that’s also equal to the sine

of theta over C. And the proof of adding

this here is identical. We’ve picked B arbitrarily, B

as a side, we could have done the exact same thing with theta

and C, but instead of dropping the altitude here, we would

have had to drop one of the other altitudes. And 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 — 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 one side and its 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. If you know three of these

things, you can figure out the fourth. And that’s what’s useful

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.

mephatboiPost authori only wanted to see how u get the third part, sin C/c

but that's the part u skipped!

Has SPost authorthis is easy stuff but it would be much easier to explain if you used values

mykeuserPost authorExcellent!

I memorised this, its the same:

sin(alpha)/sin(beta) = A/B

Daniel Loreto MartínezPost authorActually, transitivity is not a law, is an axiom, so it CANNOT be proved, its a statement which needs no demonstration, it is simply true. Answering mephatboi, the process for getting sine gamma over C (or sine C/c) is the same, but in this case you are going to draw a perpendicular line

to B that passes through beta.

Daniel Loreto MartínezPost authorAt 00:49, the triangle has just one angle without name, and this is going to be gamma. So when you draw the perpendicular line to B you will have the same situation than in the video, rather instead of B and alpha you'll have gamma and A. Good luck! (this post is the following part from the former)

ppardeePost authorOpposite and adjacent are always relative to theta. The 90 degree angle should always be opposite the hypotenuse (I believe)

ppardeePost authorMore accurately, the opposite and adjacent sides are always relative to the angle you are solving for (or with). The hypotenuse is the longest side of a right triangle.

Ada AdaaPost authorHow would that aid the proof?

spudcole319Post authorJust a question, i may not know much about trigonometry, but what's wrong with using two small right triangles to prove the law of sines? What was inconsistent?

Also, how would you prove the law of sines? because i really do wanna learn about trigonometry and to see a different way of proving this would be really helpful.

PaulorificPost authorNothing, using the law is just faster.

seandavidrPost authorCould you show how to use the law of sines to solve physics problems involving forces and vectors.

dachu108Post authorThank you, I finally know how the equation came to be now :]

Eileen BrayPost authorHe's dividing both sides by A and B so sin(alpha) and A are together on one side of the equation and sin(beta) and B are together on the other side. A and alpha are like partners, B and beta are like partners. They belong together.

Matt OgrodzinskiPost authorhe's bringing the equation down to the law of sines as seen, well everywhere you see the law of sines.

SolamPost authorur a good man

Moonlightkitteh101Post authorThank you so much: you have just saved my math grade!

hobomnkyPost authorily

MartianSporePost author@KhroniclesOfNothing I forgot that if there is no need to prove something that there is no reason to know something. Some people actually enjoy math and are interested in these things. Other people, learn better when they can understand why things are.

red115dragonPost author@PTL0

now THAT is True!!! 😉

<3 KA

Word Sailor ADD INFUNITEMSPost authorNice job! The Law of Sines proof can take you to a higher trickynometric place. It's definitely NOT the height of absurdity although ALTITUDE is certainly involved. Thanks for a colorful and clear presentation. On a less goofy note, the relationship shown is an interesting one.

Daniel LittlewoodPost authorThis video just put me 2 weeks ahead in my maths class xD

the GhostPost authoryou saved my life there 😀

Mark Felix AdulPost authorthanks for teaching i have many troubles in math especially in trigo.

sccr2009Post author@TheNevikProject You're right, I've noticed it doesnt work in all triangles, its just hard finding those, or finding a relationship in those where it doesnt work.

Daniel BrighamPost authorour math teacher taught us "Oscar Has A Hairy Old Ass" Opposite hypotenuse, adjacent hypotenuse, and opposite adjacent. I think I like SOH CAH TOA though

lpbugPost author@daymare10110 u honestly made me lol irl

ikol12Post author@daymare10110 Damn! I wish I had a teacher to teach me about Oscar's hairy old ass!

Lisa CaseyPost authorThank you – I'm in need of a trig brush up and your concise explanation is just right for me. I always need to understand the derivation of formulas in order to remember them. Your simple explanation is very helpful.

DepressedGraduatePost authorlmfao daymare XD

judo1221Post authorI wish you were my teacher

acerbic42Post authorSaw you on Colbert, congrats man.

juifuPost authorI need the proof of sine of beta over B equals to sine alpha over A equals sine theta over C "EQUALS TO 2R"? I saw that in some textbooks. thank you

meterotronicPost authordoes not help my intuition….your a very intelligent showoff.

meterotronicPost authoroh…and your favorite word is aribitrary…i've noticed that

HectorL360Post authorSome Old Hippy Caught Another Hippy Trippin On Acid.

SaiixxPost authorhey can you show the other proof using area of triangle? thanks!

Glendale TampusPost authoroh men thanks for this video it helps me alote ^^

abb bddPost authorSOH CAH TOA = Sex On Hard Concrete Always Hurts The Others Ass

Sanjay SinghPost authorGreat Video

krystian1333Post authorThis video explains nothing. So how do you do you solve the problem?

friscomelt4lifePost authorwhy am i paying my stupid math teacher 600$ when i get this amazingness for free? fuck universities what a waste of god damn money.

Chris ParkPost authorthx man,,,i was boutta fail my test lik real top.

WakeUp OrDieTryingPost authorwhat idiots disliked this video ?

it amazes me..

Lee DavidPost authorAre you greek? You are using the greek alphabet. (alpha and beta)

thbrightdayPost authorIn the beginning, there was the 4 mathematical functions: addition, subtraction, multiplication, and division. These 4 all lived in peace and harmony. But then, a great evil changed the mathematical world forever: X and Y. These letters attacked the peaceful world of the math functions, and the rest of the letter followed, leaving destruction and devastation in their wake.

Only the Avatar, who mastered all 4 function could save us, but he disappeared.

We most find a new Avatar to save our world!

johnson MannPost authoryou asian, you shouldnt be failing any math test

Reptilian5491Post authoris that.. MS paint??

Arnold YuPost authorshoosh *pap*

Muskaan DudejaPost authorI have been on a trigonometry marathon for the last three hours and my head hurts but the videos keep coming because of the playlist and essentially I can do nothing but watch.

LostStarzOfTheSkyPost authorI was taught Some Old Hobo (Sine Opposite/Hypotenuse) Caught Another Hobo (Cosine Adjacent/Hypo) Tripping on Acid (Tan Opposite/ Adjacent) SohCahToa sounds like I'm speaking a different language

qqqPost authorscience uses greek letters to signify certain variables.

Casper BmanPost authori remember Saint Oliver's Horse Came Ambeling Home To Oliver's Aunt.

What is this software

David HdzPost authorThanks Sal, you never fail me

LaurelindoPost authorThis is exactly how I think you are supposed to learn math – try to understand the exact reasonings behind things, so that you can derive them yourself on the spot.

Donkey KongPost authorWhy do you divide the equation; is there a reason?

The Bacon WarriorPost authorthank you

antonio bortoniPost authorI have been triying to find this explanation everywere… thanks.

ultimate piratePost authorpewdiepie has 40 million subs

khan has 3 million

is it only me that thinks this is a little unfair?

Pᴀʀᴀsᴇʟᴇɴᴇ TᴀᴏPost authorAs soon I saw the triangle being drawn by MS Paint-like lines, I knew this was an old video!

2007? That's near the beginning of Khan Academy. I was still a Freshman in high school when this video was made… wow.

satisfictionPost author4:02 good ole Transitive Property of Equality

Manoharan PGTPost authorvery nice

Genius by DesignPost authorWhat happen when C is greater than 90 ?

Jerferson de MatosPost authorCan I figure out something in a non-right triangle that only gives me the area and the sum of the angles? I'm really wanting to know

saurav kumarPost authorThanks sir…

Omkar GharatPost authorThis is sine rule, am i right?

Martin de SalterainPost author"Hopefully that relationship will be the Law of Sines. Otherwise I would have to rename this video." LOL

Pollen ApplebeePost authorThis proof is somewhat disappointing because it only works on acute triangles. If he'd have drawn a triangle with an obtuse angle there would've been extra steps necessary, so the proof is not actually generalized. You can't draw an altitude resulting in a right triangle from the non-obtuse angles in that case.

Catalin MPost authorwhat about the proof of soh cah toa?

Rose BPost authorjust to clarify to proove c/sineC you draw a horizontal line across angle C and that would be h/a or or h/b?

Kamran shafiPost authorI was confused when i saw this law in physics……but now i understood it completely…….thank u