Proof of the law of cosines | Trig identities and examples | Trigonometry | Khan Academy

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Proof of the law of cosines to solve for a side of a triangle when 2 sides and an angle are known.
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[@KUDANTOURAGE]

The cosine rule is the full version of pythagors' theorem......that second part of the formula is always left out because in a right angled triangle , the cosine is zero hope someone finds this helpful :D

[@teemo8247]

ahhh that makes snese

[@KUDANTOURAGE]

@@teemo8247 thanks waited 12 years for someone to acknowledge it too

[@rockysmith6105]

Your comment would apply much more to something like the Bretschneider's quadrilateral area formula being the so called complete version of Brahmagumpta's cyclic quadrilateral area formula because of the non-cyclic adaption term. However, in the case of the law of cosines as your last point states sure h^2=a^2+b^2-2ab*cos(pi/2) is in fact just h^2=a^2+b^2, but the important thing to bare in mind is that suppose you choose an angle t where the hypotenuse is adjacent annnnnd... side a arbitrarily, then b^2=a^2+h^2-2ah*cos(t) which isn't so arbitrary, and yet still applies to a right triangle. Yes, this isn't as simple as just saying b^2=h^2-a^2, and technically h*cos(t) is just a in trigonometric disguise so they are infact easy to show as equivalent, but that's not my point. My point really is that the pythagorean theorem isn't just some formula, nor is the law of cosines it's complete form or anything of that sort- I just find that logic completely disregards the pythagorean theorem as it's own entity and identity which is in my mind not alright. Now... if you re-phrase and say that the pythagorean theorem is a special case, even an identity, of the law of cosines for when you are finding the hypotenuse of a right triangle I'd say sure. The pythagorean theorem is in no way incomplete, as it never was intended to apply to any triangle except the very, VERY special family of right triangles. If this wasn't too tedious, I hope you found this helpful. I... don't mean to be harsh I just don't like it when things as fantastic as the pythagorean theorem get thrown out with the rest of the whole world of formulae... and the LAW of cosines for that matter. If you can't tell, I'm also one of the sticklers that says NEVER forget +c when you are doing an anti-derivative, and NO, an integral sign without bounds is NOT a sign saying "integrate the expression" it says to anti-differentiate the expression... and now I'm rambling, ranting, and raving. I hope there was something in here you found useful... if it wasn't too tedious or offensive. I mention Brahmagumpta and Bretschneider's area formulae because they are shown in Mathologer's most recent video on 1+2+3=1*2*3, the 3,4,5 triangle, and Heron's formula which is some awesome stuff I recommend you take a gander of. Good day!

[@teemo8247]

@@rockysmith6105 why did you write so much to a youtub comment

[@YannickHollanda]

That's so cool, I never realised. Thanks to you my Trig knowledge has broadened. You're 13 yr old comment helped me prepare for my upcoming Entrance exam, I hope you know that 🙏

[@zeekfromthecreek]

"I'll be impressed if you're still doing trig when you're 40." I'm 59. Catching up on missed math.

[@crow2533]

Andre Duval you are an absolute maldad my friend.

[@PranabMallick.]

Wow And I am just 14

[@hanniballecter3727]

@@PranabMallick. same homie (I never sleep cuz sleep is the cousin of death)

[@cxllmegeo3557]

@@hanniballecter3727 ;-;

[@purpose2591]

bro what

[@MrJdcirbo]

This is really interesting. The law of cosines is the Pythagorean Theorem minus a correctional factor of 2cbcos(angle). In the special case of a right triangle where side 'a' is the hypotenuse and the angle is 90°, you get a correctional factor of 2cbcos(90°) which is zero. That's pretty useful

[@nanosanchezlopez575]

I love how shocked you were in the begining of the video when you realized the red line you were making wasn't straight lol

[@SoulofAnotherDeity]

@8:50 - "I'm not a big fan of just memorizing it without knowing where it came from" I'm the same when it comes to learning pretty much _anything_. :)

[@iliekmathphysics]

I agree I get so ma like OMG WHY DO I FO THAT STOP JT

[@hunterepicness4303]

@@iliekmathphysics what?

[@demonetizedhistory5806]

It's so crazy that this dude can do trigonometry with only variables and fully explain a situation with no numbers and that's completely unknown. I'm really astonished.

[@eduardoandrescontrerasrome6703]

That's the power AND beauty of math IMO.

[@rockysmith6105]

@@eduardoandrescontrerasrome6703 Yes, yes it sure is👍

[@khanacademy]

Solve for angle B (either with law of sines or law of cosines), then you will know angle A since A+B+C=180, then you could solve for a.

[@tiggerbear3]

Hey, I'm 40 and working on the laws of sine and cosine...should I be insulted here?? lol j/k :)

[@hunterepicness4303]

*Tangents have left the chat*

[@mishaoberoie159]

no...you are welcome here

[@smackdashitoutchu7755]

POV: The commentator was 40 when he wrote this comment. As of today in 2021, he is 47.

[@tiggerbear3]

@@smackdashitoutchu7755 😂 yeah, but totally remember this vid...😎😉

[@Fictionarious]

Thanks Sal, I was embarking into tall calculus 3 grass when a wild geometrical statement appeared! I couldn't catch it until after watching this.

[@chuckjls]

I am sixty and I am still doing trigonometry. This stuff is for life. Ok, the law of cosines is the basis for vector analysis, specifically the dot product of two vectors, which you learn about in Calculus III. Thanks Khan for the quick refresher.

[@MoonLight11023]

Law of cosines can be proven by drawing a triangle abc in Cartesian coordinates in which ab side = A, bc side = B, and ca side = C. The triangle points are as follows: a(0,0), b(A,0) and c(Ccos(x),Csin(x)) where x is the angle between A and C sides. Using distance formula: B^2 = (A-Ccos(x))^2 + (0-Csin(x))^2. B^2 = A^2-2ACcos(x)+C^2.[cos(x)]^2+C^2.[sin(x)]^2. B^2 = A^2+C^2-2ACcos(x).

[@shellyeditsalot]

You really helped! I don't need to study from my textbook now =D!

[@adityabhandari271]

thank you so much sal, i know im writing this 15 years after this video was released but these videos have helped me so much i really appreciate you doing this so tysm!

[@vister6757]

Thanks! Your explanation is perfectly clear to me, and that helps me to remember it.

[@committedcoder3352]

can you redo this with your now-neat-hand writing?

[@MENNENGER]

Learning in my point of view should be fun too. This guy is so much fun. I love it. Thanks KHAN.

[@gdogvibes1]

very neat!

[@user-df3cd1wo2b]

So good and its simple than l thought

[@hunterepicness4303]

I understand *Perfectly* . Thank You.

[@mwerensteijn]

Very clear explained!

[@saraha.k.861]

your amazing! And you actually made this interesting too!

[@MuletTheGreat]

How did I become a programmer? I have trouble with math, yet I make video games for a living. -.-

[@saudinho1436]

Thanks for this intersting proof

[@ankitaaarya]

Its amazing that physicists and mathematicians solved made and discovered new tools for their convenient now every physicist and mathematician uses each other's tools for their convenience.

[@demonking1101]

your methods of teaching are just beautiful and here i am getting depressed due to my math teacher

[@aseelso5794]

Thank you perfect 😊

[@TWRHRS]

You came a long way Sal lmao

[@Dr.Nitmolol]

that was really helpful ............ proving it, rather then memorizing it.........sal

[@horatiohornblowerer]

That might what be his concise and helpful explanations of elegant properties.

[@fahmidaafreen9855]

Thank you so much, really.

[@Mr707derby]

I'm forty and doing trigonometry...Ty

[@PreCalculusRocks]

Thanks so much!! I love trig :)

[@zeetee6]

this is hard but he makes it look easy !!!

[@fredflintstone8048]

I'm 65 and I'm still using trig. I use it in computer software that I write for control system, and I use it even more in my machine shop.

[@prakh1250]

2:01 *Top 5 Moments You Won't Believe*

[@khushikanodia1899]

thanks mate that helped

[@leonvdd]

Yes wow thx this is so great now I finally know how to do this :-)

[@marleyluber]

thanks so much!!! my teacher is allowing us 3 pts. of extra credit to memorize this proof! thanks!!!!!

[@seriousmax]

A mess of pixels.

[@DogeFrom2014]

Max Webster Uploaded on 2007, what did you expect?

[@badymojoy]

+DOGE™ I expect an update.

[@DogeFrom2014]

badymojoy Valid point, but Khan has many other new topics to teach than reteaching trigonometry. Although you could ask him for an updated version, but the fact that no one is complaining means that Khan is okay not reteaching it.

[@badymojoy]

+DOGE™ From what I've noticed, there are a small portion of videos on KA still using this low resolution, where things like equal signs look like c's. However small the minority, I consider myself part of those desiring to see an update of those videos--whenever Khan or someone else at KA's time permits.

[@mrbartinio]

still waiting

[@powerslayer9069]

thanks!!

[@jugglingtheapples]

OH my god that was messy. I thought I was using the line tool

[@MrMarkgyuro]

thank you! I appreciate your videos! :)

[@duplitoaejr]

Yeah, I would be impressed too if someone (who is not a math teacher) knows trigonometry after turning 40. Most of the math terms and equations I learned in high school are pretty much forgotten.

[@vivialord5260]

Thanks

[@Paraselene_Tao]

We need this video remade; it's aged quite a bit-in it's time it was difficult to follow too.

[@newtonsharma]

thank you very much.

[@gurumoorthy8247]

Wonderful

[@over-rakum]

I'm so thank you

[@kevindave5850]

The older I grow the more sensible mathematics becomes for me to understand.

[@m22d52]

brilliant

[@peachboy595]

8:43 LOL I have a test tomorrow and I haven't learnt this..

[@youcanteachthem]

I am in my late 40's and I am doing trigs :) I can't memorize, though!

[@moh3921]

for the Obtuse Angle in Non-right angle triangle. shall we take the cosine of that angle when we want to calculate the length of the line opposite to that angle, or we take the cosine for the supplementary angle?

[@QuranWithUrduTranslation]

Some People Have = Sin P / H Curly brown hairs = Cos B \ H They use proper Brush = Tan P \ B

[@randomazed]

I'm doing trig at 40. Gonna take that one as a compliment. :)

[@Yu2Kal]

Yea this guy is correct, I got the same answers as him, also that isn't a cosine rule question, you just need to use the sine rule twice and use sum basic geometry facts.

[@JeniferSheppard]

I wish you were teaching my math classes at Uni..

[@janenikiita7919]

u helped my lord of Cosine

[@EXPComics]

xD now I feel perpared for this Quiz LOL

[@MyCarlover]

can you make a video where you derive sine and cosine law in obtuse triangles?... plzzzzz

[@juliaflores6935]

superb

[@debashishpradhan2325]

I don't understand why khanacademy app is free.. Perhaps Its the cause why its not popular

[@markgreen6229]

Your videos are so great but this one sounds like you woke up with a hangover and was asked to prove something.

[@nicholasmartin6353]

Beautiful

[@cbsegm]

I'm 57 and still doing Trig. Calc next year.

[@TheWeeHenry]

+Carl Silverstein I'm 57 too and still using it.

[@Dman82499]

?

[@malcolmfrater7029]

+Dman82499.....I use it in my job as a pipefitter to calculate angles and lengths of "travel" when they're not the standard 45 degrees.

[@chair1694]

Man, I love ya

[@angiebaby4292]

hi i have a question ok im doin the lows of cosine nd the worksheet is asking me to find AB where they gave me little a nd b which is a-12 nd b- 28 nd angle C= 26 now my question is which one of the formulas should i use to get side c nd also AB?

[@caileanjones9713]

Love it

[@jiwonkim7536]

wow that video is so old.

[@Francesco_Armillotta]

can I ask what software do you use? and overall it's free?

[@guestdeej]

hello.. im just curios with your question that's why i answered it.. but i dont know if its correct. anyway, here's my answer.. a=12.52 A=74.56(i got it when i subtrct 180 form 38 deg and 67.44 deg.) what do you think?? ;D am i correct??

[@xzxz619]

And such what might that be?

[@dackid2831]

God I love algebra!

[@smoothieGurl23]

I'm confused. I think I need to use this method for my homework except I don't quite understand. If I have a=8 and c=10. I even have two of the interior angles but I'm lost on how to begin. Can you please help!?

[@MARSTVCHANNEL]

Could someone explain the origins of the calculations at 6:02, or where I can find that particular Khan video where it is explained? I'm 40 and can't remember!

[@Zedriodor]

***** Quadratics.

[@MARSTVCHANNEL]

Where does the 2cbcosØ come from?

[@Zedriodor]

When you expand any square like this (a-b)^2 it becomes (a-b)x(a-b) which you then use FOIL or any other method to multiply it out such that it gives you this: a*a - a*b + b*b - b*a. Which simplifies to: a^2 +b^2 -2ab.

[@MARSTVCHANNEL]

I understand that, but how does this relate to my question? This occurs at 6:17 btw.

[@MARSTVCHANNEL]

a^2 =

[@horatiohornblowerer]

There is something beautiful here.

[@vmiget01]

so colorful! xD

[@aarongould3538]

i love you sal khan

[@soccerbabez45]

I need to know how to solve and draw out diagrams using te formula for law of cosine :(

[@peacedudepeace]

the first one or the two after?

[@sdH2Opolo]

i think it would have been easier to understand if you would have just started with a right triangle

[@Djcrzy]

im not sure but i think cos law is only for SAS and SSS (side angle side and side side side)

[@abichan210]

i love watching derivations asdfghjkl

[@strike-pw571]

SLACKER!!!!111!!1one!

[@MartianSpore]

@Awlaad1 If you only know three angles of a triangle you do not have enough information to solve for the triangle.

[@eipimath]

Thanks for sharing! I posted a short video on deriving the law of cosines, It should apply for all angles (acute, obtuse, reflex, negative). Hope to get your thoughts.

[@scriptblaster4237]

*analyzing* 86% complete.

[@ankitaaarya]

😂😂😂

[@jonf6509]

Im 36 and i had to come back to this crap now haha Why wouldnt one be doing trig when their 40?

[@ARN48411]

OMG, This 14 years ago video.. It's 2022

[@BigPokemonFan987]

If ur in trigonometry lol i have to do this in geometry.

[@footballherocjp]

LOL ur gunna be using this for the next couple of years bud, guarantee you see it again. You still have ways to go.

[@BigPokemonFan987]

Lord Jean-Pierre haha its hard stuff

[@loganbmoy]

HarmlessGirlscout Lol I a m doing this in 7 grade

[@MrMistery101]

How many of us are watching this on the day before the test/quiz?

[@charlesrambo7845]

What about when theta is an obtuse angle?

[@PancakeDoesGaming]

+Charles Rambo Then cos theta would be negative, but there won't be problems.

[@jiyaali3840]

Sal khan be like lemme change the color of the pen!!!

[@drudru5992]

Need a better black board. Lost me at the very end where the writing became difficult to make out.

[@semoge3249]

Dru Dru if you could not follow what he was saying, you clearly are not use to trig and (a+b)^2

[@Nobiashi]

That only works for right triangles

[@taichingkan79]

The law of sines only works when you have an angle and the side opposite it given to you.

[@JellyIsPeng]

how many colours does he have?!?! hahaha

[@jackyoung8354]

why is everybody 40 years old?

[@peacedudepeace]

what do you mean?

[@celenamorris]

I got lost fml