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
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!
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 🙏
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
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.
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.
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).
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!
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.
+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.
I love Khan Academy, and I really appreciate all the great content you provide. If I may, I would like to ask if it’s possible to redo this video in better quality. I can barely read the text, but I’m eager to understand the material. A new video with improved clarity could be really helpful. Thank you!
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.
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.
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?
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.
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??
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!
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.
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?
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!?
Hmm... I wanted to know the proof for Law of Cosines in order to prove Pythagorean Theorem. I want proof more axiomatic to prevent this circular reasoning...
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.
@@anindaaninda3383 UA-cam prevents me from posting the direct links. You have to copy the links and replace the similar part that is already in the present URL
Was top of my class, "learned" this back in 2009, and still don't understand or like it. Probably the sole reason I'm not any sort of engineering major. Death.
![Pythagoras Would Be Proud: High School Students' New Proof of the Pythagorean Theorem [TRIGONOMETRY]](/img/n.gif)
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
ahhh that makes snese
@@teemo8247 thanks waited 12 years for someone to acknowledge it too
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!
@@rockysmith6105 why did you write so much to a youtub comment
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 🙏
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
@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_. :)
I agree I get so ma like OMG WHY DO I FO THAT STOP JT
@@iliekmathphysics what?
"I'll be impressed if you're still doing trig when you're 40." I'm 59. Catching up on missed math.
Andre Duval you are an absolute maldad my friend.
Wow
And I am just 14
@@PranabMallick. same homie (I never sleep cuz sleep is the cousin of death)
@@hanniballecter3727 ;-;
bro what
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.
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
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.
That's the power AND beauty of math IMO.
@@eduardoandrescontrerasrome6703 Yes, yes it sure is👍
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.
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).
Hey, I'm 40 and working on the laws of sine and cosine...should I be insulted here?? lol j/k :)
*Tangents have left the chat*
no...you are welcome here
POV: The commentator was 40 when he wrote this comment. As of today in 2021, he is 47.
@@smackdashitoutchu7755 😂 yeah, but totally remember this vid...😎😉
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.
0:51 bro i hope you are impressed. Doing trig and pre calculus at 48
im impressed
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!
You really helped! I don't need to study from my textbook now =D!
A mess of pixels.
Max Webster Uploaded on 2007, what did you expect?
+DOGE™ I expect an update.
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.
+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.
still waiting
Thanks! Your explanation is perfectly clear to me, and that helps me to remember it.
I love Khan Academy, and I really appreciate all the great content you provide. If I may, I would like to ask if it’s possible to redo this video in better quality. I can barely read the text, but I’m eager to understand the material. A new video with improved clarity could be really helpful. Thank you!
Learning in my point of view should be fun too. This guy is so much fun. I love it. Thanks KHAN.
2:01
*Top 5 Moments You Won't Believe*
How did I become a programmer? I have trouble with math, yet I make video games for a living. -.-
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.
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.
Some People Have = Sin P / H
Curly brown hairs = Cos B \ H
They use proper Brush = Tan P \ B
So good and its simple than l thought
your methods of teaching are just beautiful and here i am getting depressed due to my math teacher
this is hard but he makes it look easy !!!
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.
I'm forty and doing trigonometry...Ty
You came a long way Sal lmao
I understand *Perfectly* . Thank You.
that was really helpful ............ proving it, rather then memorizing it.........sal
That might what be his concise and helpful explanations of elegant properties.
I'm doing trig at 40. Gonna take that one as a compliment. :)
I have you beat by 41 years...
@CaryInVictoria heck yes, that's awesome!! 👏🏻
... and I've promptly forgotten it again.
Thanks for this intersting proof
Very clear explained!
your amazing! And you actually made this interesting too!
Thank you
perfect 😊
I'm 57 and still doing Trig. Calc next year.
+Carl Silverstein I'm 57 too and still using it.
?
+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.
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?
We need this video remade; it's aged quite a bit-in it's time it was difficult to follow too.
thanks so much!!! my teacher is allowing us 3 pts. of extra credit to memorize this proof! thanks!!!!!
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.
can you make a video where you derive sine and cosine law in obtuse triangles?... plzzzzz
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??
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!
***** Quadratics.
Where does the 2cbcosØ come from?
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.
I understand that, but how does this relate to my question? This occurs at 6:17 btw.
a^2 =
The older I grow the more sensible mathematics becomes for me to understand.
OH my god that was messy. I thought I was using the line tool
very neat!
I am in my late 40's and I am doing trigs :)
I can't memorize, though!
I wish you were teaching my math classes at Uni..
8:43 LOL I have a test tomorrow and I haven't learnt this..
Thanks so much!! I love trig :)
Yes wow thx this is so great now I finally know how to do this :-)
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?
What about when theta is an obtuse angle?
+Charles Rambo
Then cos theta would be negative, but there won't be problems.
thanks mate that helped
thank you!
I appreciate your videos! :)
Need a better black board. Lost me at the very end where the writing became difficult to make out.
Dru Dru if you could not follow what he was saying, you clearly are not use to trig and (a+b)^2
u helped my lord of Cosine
@Awlaad1 If you only know three angles of a triangle you do not have enough information to solve for the triangle.
I don't understand why khanacademy app is free.. Perhaps Its the cause why its not popular
can I ask what software do you use? and overall it's free?
*analyzing* 86% complete.
😂😂😂
thank you very much.
The law of sines only works when you have an angle and the side opposite it given to you.
wow that video is so old.
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!?
im not sure but i think cos law is only for SAS and SSS (side angle side and side side side)
Your videos are so great but this one sounds like you woke up with a hangover and was asked to prove something.
Hmm... I wanted to know the proof for Law of Cosines in order to prove Pythagorean Theorem. I want proof more axiomatic to prevent this circular reasoning...
There are thousands of proofs fro the pythagorean theorem that you can use and then use it to prove this so it is in no way circular
For*
xD now I feel perpared for this Quiz LOL
Thanks
thanks!!
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.
And such what might that be?
the first one or the two after?
I need to know how to solve and draw out diagrams using te formula for law of cosine :(
i think it would have been easier to understand if you would have just started with a right triangle
I'm so thank you
There is something beautiful here.
brilliant
If ur in trigonometry lol i have to do this in geometry.
LOL ur gunna be using this for the next couple of years bud, guarantee you see it again. You still have ways to go.
Lord Jean-Pierre haha its hard stuff
HarmlessGirlscout Lol I a m doing this in 7 grade
God I love algebra!
why is everybody 40 years old?
superb
You said C side is interesting. Are you a Squeak + Seaside programmer?
That only works for right triangles
AL KASHI, right ???
If anyone else is curious, "last video" is referring to 2 videos:
watch?v=1vamogV81Y8
watch?v=4CNnPgabrLE
Buddy, links arent there
@@anindaaninda3383 UA-cam prevents me from posting the direct links. You have to copy the links and replace the similar part that is already in the present URL
i love watching derivations asdfghjkl
6:21 i still dont understand how you got 2cbcos(theta)
+Anthony Volpe Try multiplying the term out, you will get c^2 -bcCos(thet) - cbCos(thet) + b^a. You add the two middle terms and viola
I'm watching this video in 2024, and this video was uploaded in 2007 when I was born
I''m watching it on my 17th birthday :)
8:52 "A year or two from now when you go to college and it's been four years since you took trigonometry..." A year or two... four years... 2 = 4?
he means a year or two after highschool, when ur in college. but u learn trigonometry four years BEFORE college.
Great video, just need a better quality software... The era of paint is over
Star Trek: Into Darkness = badass.
Love it
Beautiful
i love you (not literally)
I’m 17 years now and I’m also not a fan of memorizing things…
Was top of my class, "learned" this back in 2009, and still don't understand or like it.
Probably the sole reason I'm not any sort of engineering major. Death.
***** No, I lived in Texas.
teenroses Sorry, but this is the funnies thing I've read all day XD