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

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

Articles, Blog , , 70 Comments

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.

70 thoughts on “Proof: Law of sines | Trig identities and examples | Trigonometry | Khan Academy

  • mephatboi Post author

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

    but that's the part u skipped!

  • Has S Post author

    this is easy stuff but it would be much easier to explain if you used values

  • mykeuser Post author


    I memorised this, its the same:
    sin(alpha)/sin(beta) = A/B

  • Daniel Loreto Martínez Post author

    Actually, 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 Martínez Post author

    At 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)

  • ppardee Post author

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

  • ppardee Post author

    More 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 Adaa Post author

    How would that aid the proof?

  • spudcole319 Post author

    Just 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.

  • Paulorific Post author

    Nothing, using the law is just faster.

  • seandavidr Post author

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

  • dachu108 Post author

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

  • Eileen Bray Post author

    He'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 Ogrodzinski Post author

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

  • Solam Post author

    ur a good man

  • Moonlightkitteh101 Post author

    Thank you so much: you have just saved my math grade!

  • hobomnky Post author


  • MartianSpore Post 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.

  • red115dragon Post author

    now THAT is True!!! 😉
    <3 KA

  • Word Sailor ADD INFUNITEMS Post author

    Nice 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 Littlewood Post author

    This video just put me 2 weeks ahead in my maths class xD

  • the Ghost Post author

    you saved my life there 😀

  • Mark Felix Adul Post author

    thanks for teaching i have many troubles in math especially in trigo.

  • sccr2009 Post 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 Brigham Post author

    our 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

  • lpbug Post author

    @daymare10110 u honestly made me lol irl

  • ikol12 Post author

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

  • Lisa Casey Post author

    Thank 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.

  • DepressedGraduate Post author

    lmfao daymare XD

  • judo1221 Post author

    I wish you were my teacher

  • acerbic42 Post author

    Saw you on Colbert, congrats man.

  • juifu Post author

    I 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

  • meterotronic Post author

    does not help my intuition….your a very intelligent showoff.

  • meterotronic Post author

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

  • HectorL360 Post author

    Some Old Hippy Caught Another Hippy Trippin On Acid.

  • Saiixx Post author

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

  • Glendale Tampus Post author

    oh men thanks for this video it helps me alote ^^

  • abb bdd Post author

    SOH CAH TOA = Sex On Hard Concrete Always Hurts The Others Ass

  • Sanjay Singh Post author

    Great Video

  • krystian1333 Post author

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

  • friscomelt4life Post author

    why 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 Park Post author

    thx man,,,i was boutta fail my test lik real top.

  • WakeUp OrDieTrying Post author

    what idiots disliked this video ?

    it amazes me..

  • Lee David Post author

    Are you greek? You are using the greek alphabet. (alpha and beta)

  • thbrightday Post author

    In 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 Mann Post author

    you asian, you shouldnt be failing any math test

  • Reptilian5491 Post author

    is that.. MS paint??

  • Arnold Yu Post author

    shoosh *pap*

  • Muskaan Dudeja Post author

    I 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.

  • LostStarzOfTheSky Post author

    I 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

  • qqq Post author

    science uses greek letters to signify certain variables.

  • Casper Bman Post author

    i remember Saint Oliver's Horse Came Ambeling Home To Oliver's Aunt.
    What is this software

  • David Hdz Post author

    Thanks Sal, you never fail me

  • Laurelindo Post author

    This 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 Kong Post author

    Why do you divide the equation; is there a reason?

  • The Bacon Warrior Post author

    thank you

  • antonio bortoni Post author

    I have been triying to find this explanation everywere… thanks.

  • ultimate pirate Post author

    pewdiepie has 40 million subs
    khan has 3 million
    is it only me that thinks this is a little unfair?

  • Pᴀʀᴀsᴇʟᴇɴᴇ Tᴀᴏ Post author

    As 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.

  • satisfiction Post author

    4:02 good ole Transitive Property of Equality

  • Manoharan PGT Post author

    very nice

  • Genius by Design Post author

    What happen when C is greater than 90 ?

  • Jerferson de Matos Post author

    Can 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 kumar Post author

    Thanks sir…

  • Omkar Gharat Post author

    This is sine rule, am i right?

  • Martin de Salterain Post author

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

  • Pollen Applebee Post author

    This 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 M Post author

    what about the proof of soh cah toa?

  • Rose B Post author

    just 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 shafi Post author

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

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