Sine and Cosine Addition Formula Proof

[@dragoscalin4883]

Da, este cea mai vehiculată demonstrație geometrică cu ajutorul geometriei plane, a relațiilor trigonometrice dintre suma a două unghiuri, respectiv sinusul și cosinusul sumei respectiv diferenței a două unghiuri, utilizând cercul unitate(cercul trigonometric), găsindu-se și în manualele de matematică. Mulțumim, vă doresc sănătate și succes în continuare.

[@somethingkabir936]

The best and most easy and logical proof!!! Thank you 😊

[@biancaselvarajah2316]

i agree.

[@newwavenewwave1035]

It's great to finally see a video that demonstrates several trig formulas in one diagram only. Not even to mention that if one masters that he can hence easily get a great bunch of related trig identities

[@jackflash8756]

I agree this is the best proof I've seen on you-tube. When people draw circles with unit 1 radius to recreate that same digram, you 'can't see the woods for the trees' . You drew one triangle on top of the other and all of a sudden it is clear as day.

[@finpas9915]

How clear and beautiful. Thank you!

[@raminrasouli191]

This was the best video I have seen about this proof. Thank you.

[@descendantt]

It's the simpliest proof I've ever seen. Thank you so much!

[@Takatsu_Jie]

I have a good maths teacher but his explanation was really confusing but your maths vid broke it down into tiny steps which made it so much easier to digest thank you👍

[@samakolBanbol]

I don't care about proofs, but I needed to understand visually why adding or subtracting angle ratios is done thusly and I looked at the proof somewhere else but couldn't understand it... Now I understand exctly and what's happening and I don't even need to memorize anything about compound angles.. I can draw it on the spot and derive what I need even 10 years from now!! Thank you Erik!

[@redfinance3403]

Very good Proof, thank you. I prefer this one rather than the one will a lot of fractions and assumptions, this one is much cleaner and well defined.

[@MohitSharma-gp2ht]

This channel deserve millions of followers

[@eniangeniang3703]

this guy is good. big ups man. this is the simplest proof i've ever seen on this topic

[@lalityt07]

That's the best intuitive proof I have found yet. Great job 👍🏻

[@MillValleyBoczeks]

Hey, you were that smart kid in my Calculus 101 class in Ithaca 40 years ago. I see that you found your calling! Well done.

[@erikthered109]

That's me!

[@fomalhautXP]

If there are talent of teaching, this is what it is. Thank you so much for easy explanation.

[@frankhong6313]

Wow, that's hella intuitive. So intuitive that it was forever written into my gene!

[@Sam-fq1ho]

Completely brilliant. Thank you so much!

[@saidndimbwa4976]

This is easiest way of this proof...you are genuine! Live long!

[@sajidrafique375]

I went through 4-5 videos ...Yours is BEST

[@priscakembo]

This guy is great

[@TheALEXMOTO]

That's the best explanation I've seen in years.

[@103vaishnavi]

really simple diagram-much easier to understand than other diagrams-thank you

[@NickForrer]

Very clear and concise - thank you!

[@FizOlimp]

That's so beautifull proof! I thank you for this explanation. I haven't found this evidence in my native language. 🤗

[@ShivBhardwaj9973]

One of the best video on this Topic ❤❤ Love from india 🇮🇳❤️

[@simrannahar8262]

Sir this is a very, and I mean very sophisticated proof that's been made so easy to understand, I thank ye

[@rnd_penguin]

As someone who was struggling to figure out this shit, this video is a godsend for me.

[@michaelgebremeskel6622]

You made my life easier. Thank you.

[@dVPulse]

This is fine for 0 < A+B < 90 degrees, but what about obtuse angles when we have to use the circle definitions for trig functions. Do we need another proof or is this one enough?

[@erikthered109]

First I would just remark that this proof by diagram works as long as both A and B are acute, even if 90 < A+B < 180. Next, if you have either or both A and B greater than 90, you can define A' = A - 90n and B' = B - 90m to get A+B = A'+B' + 90(n+m) where A' and B' are both acute and n+m = 1, 2, or 3. (Any multiples of 360 can be subtracted out since the sine and cosine won't change.) Now, sin(C+90) = cos(C), sin(C+180)=-sin(C), and sin(C+270)=-cos(C) and similarly for cosine. (You can verify these by coordinate geometry without having to use the addition formulas.) So, we've reduced the problem of sin(A+B) to -sin(A'+B') or ±cos(A'+B') for which the proof works. Definitely messier, yes. But I don't think you have to rely on a completely different proof.

[@9WEAVER9]

@@erikthered109 I am glad the proof works out to the appropriate result but I think it needs to be justified why you can place the top triangle on the hypotenuse of the bottom, it's just not clearly justified in your video why the hypotenuse hit the lower triangle should equal cosine of B, if both of these triangles are on the unit circle they should share a hypotenuse of one. so if that middle line was equal to one that would be clear, I'm just confused on this, but as I said the result works out so clearly I'm missing something I just wish you had explained the diagrammatic construction but then again I'm no expert on math or making videos

[@erikthered109]

Hi Anthony, thank you for your comment. I think I can understand why there might be some confusion. Only the top triangle has a hypotenuse of 1; the end of the lower triangle's hypotenuse is not on the unit circle. You may be wondering: given any right triangle on the bottom, how can I then draw a triangle with a hypotenuse of 1 on top of it, and the answer is, I can't always do that, unless the hypotenuse of the lower triangle is less than 1 unit in length. But, if the bottom triangle is too big, I can always scale down the entire diagram until the upper hypotenuse is 1; the new triangles are similar and the angles remain the same. I hope that helps clear things up.

[@c.s.842]

Wonderfull proof wonderfully explained. Thanks

[@fordtimelord8673]

Just a basic knowledge of complex numbers and Euler’s formula makes this proof almost trivial. But fascinating see the traditional real method.

[@ahmedsial382]

Your the best maths teacher ever

[@sakindrapurbe5771]

really exceptional n can be a very effective methods all thanks to ERIK. best wishes from NEPAL

[@tinkerman8527]

this is a very clever proof thank you

[@DaniloSouzaMoraes]

If we're just adding angles, the original and final x,y should be at the same distance from the origin, right? But the second triangle has a hipotenuse of 1 while the first has a hipotenuse of 1cos. Why is that?

[@9WEAVER9]

really glad I'm not the only one questioning this because the proof ends up working out to the appropriate answer but I also don't understand why cosine of B is the hypotenuse to the lower triangle, if the lower triangle is on the unit circle it's hypotenuse should be equal to one

[@AlFredo-sx2yy]

the reason why is the following. Imagine you want to use this to find the rotation of a vector. Lets say that the vector is the line that the guy from the video said has length cos(B). Forget about cos(B) for now. Lets say that this vector has a length of 69 for example. The vector has an angle of A degrees in respect to the X axis. What we want to find now is what the vector will be if we rotate it by B degrees until we reach an angle of A+B degrees. So, if you look at the unit circle (circle of radius 1) then we can start working from that line that has a length of 1 in this guy's video. What we are looking for when we draw the line that is perpendicular to the line with length 1 is a line that will cut with our original vector which had a lenght of 69 units. The point in which it cuts is obviously shorter than 69 units. What will be the length of the segment that we have from the origin all the way to the point where the cut happens? well, since we're working on the unit circle so that the second line has a length of 1, then the cut happens at a length of cos(B). That is why that line has that length, because it doesnt actually reach all the way to the edge of the unit circle. If we were working with a line of any other length, then the cut would happen at a different point. Imagine that the length was h, then the cut would happen at h*cos(B) which would scale the rest of the operations done in this video by h, which doesnt affect our final result but it makes working with those values more cumbersome until we reach the answer, which is why he chooses to use a length of 1. Hope this wall of text was somewhat understandable and sort of made sense.

[@haticeavsar4704]

Thank you so much it is so clear and easy to understand

[@bobbyearayil]

Simple and very clear. Appreciated

[@SivaKumar-AoT]

best and most easy proof !

[@DhanalaxmiMendadala]

Thank you, Mind-blowing explaination,very clear 😊

[@agby945]

The best video, thank you!

[@biancaselvarajah2316]

super video! thumbs up.

[@biancaselvarajah2316]

thank you for sharing.

[@brin6494]

This is brilliant, thanks so much!

[@冇人知我名]

wonderful proof, thank you.

[@AntonioHernandez-re9xj]

Perfect explanation!

[@Amzzfr]

omg thank you all the other videos complicate it so much!!

[@barathwinmaster8637]

Ultimate clear explanation

[@holygroovearena]

Why is the hypoteneuse of the upper right triangle 1? If it's cos it's from the unit circle, why isn't the hypoteneuse of the lower triangle also 1? Cos it's the same radius of 1

[@BA.enjoyer]

great job! I appreciate it.

[@sarahkaveh7739]

Brilliant🍒💚

[@detectiveandspynovels7140]

Fantastic ,

[@rohinarora5900]

Really helpful

[@andrewoharaaidoo8139]

God bless your soul

[@clashgaming238]

Thanks for this wonderful video sir

[@thomasbates9189]

Very helpful! Thank you

[@keeposcillating]

Excellent

[@salmancem]

Very nice

[@zeyads.el-gendy4227]

Clear, brilliant.

[@baijnathgupta8496]

Nice explanation ...thank you sir❤❤

[@gouthamraj7449]

Thank you so much for this awesome video!💯👍🏻

[@jerryjoseph3268]

Fantastic...

[@elma2410]

Thank you so much!

[@sirishchandputla]

Very nice.Thank you

[@qodirjondadaboyev8169]

Good job . Thank you

[@yunxiaaang]

I feel illuminated, thank you sir😁

[@jayasrichakravarthy320]

Thank u so much.. its very clear..... Awesome 👍

[@sangurai3789]

Ooooh thankyou King 🙏🤝🤝just Subscribed❤❤❤

[@thecheem3764]

It was really cool, THANKS

[@russellmanning7023]

Is this an actual proof though? Or does it only apply in this specific instance where ( kind of arbitrarily, and very conveniently to suit the ‘proof’) there is this weird shape of 2 triangles. One placed on top of the other, so one triangle’s side is the others’ hypotenuse. What if I were to devise some strange combination of shapes. Would it still work?

[@overlordprincekhan]

There is a saying that Mathematics is not a hard subject. It is you who is taught by an idiot teacher. And today I found this proof in video. 3 weeks straight I couldn’t figure out this but this single video has eradicated all of the doubts and allowed me to fully underatand this. Thanks

[@BaseerAcademy]

Thanks a million!

[@roygreen8265]

just asking, how called the program u are using in this video?

[@gottadomor7438]

Math proofs - the ones I can comprehend anyway ;-) - beautiful; and t/y for this one.

[@indianancientsage]

Thank you sir. So clear and beautiful..

[@15ixddivyanshudayal51]

Can we use this proof in our examinations

[@idolgin776]

Very nice!

[@jacek7178]

really good and short and clear

[@MrWick554]

Thank you so much my friend.

[@joshthedog620]

we love you Dr. J

[@Sneha-wt6yk]

Thanks a lot❤🙏

[@elissacrabtree1583]

Great proof. What app are you using?

[@erikthered109]

I'm using an Android app called LectureNotes. The app is a bit wonky but is great for this kind of video.

[@osbornlee641]

Love it!

[@humayrakadir5853]

Thank you so much this is amazing

[@dannyjohndyer]

excellent!

[@RampraveshYadav-n7v]

Thanks 🙏🙏

[@greatanimals3212]

Thank you very much sir ❤

[@gracekgosana9123]

Thank you this is helpful

[@taljune142010]

Thanks a lot!

[@marinafarouk3271]

That was really easy thanks, but I just have a question. What if the hypotenuse was NOT 1. How would we prove it ?

[@erikthered109]

Hi Marina, thanks for your question, it is a good one. The reason I can choose 1 as the length of the hypotenuse for the proof is because of similar triangles. If I make the hypotenuse length c, and keep the angles A and B the same, I get new similar triangles whose sides are now all multiplied in length by c. When I calculate sines and cosines, the factors of c divide out, and you get the same results for sin(A+B) and cos(A+B).

[@kurchak]

Damn he's good.

[@philominaduesdieker2430]

Elegant.

[@VishwasSingh-r7w]

It's very easy trick😊😊

[@ksk_futuremvrcreator]

Awesome.

[@Omar-jv6tu]

thank you sir. ➕♾ you made my day.

[@hiutungcheng489]

thank you so much

[@eliasbushisso1742]

thanks a lot

[@karnan5877]

Thank you

[@expertise0801]

So why hypotenuse of A is not 1

[@ChrisandusOdhiambo]

The best

[@LwEd-be4mv]

THANJ YOU SO MCH !!!!!!

[@RAJESHSK-ch9uc]

Thanks a lot sir