How to prove the sum-to-product identity for sine: sinA+sinB=2sin((A+B)/2)cos((A-B)/2)

[@blackpenredpen]

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[@shrihari154]

Sir now Ill completely forget memorizing these formula Because from Now on I can derive these formula anytime , anywhere Long Live BlackPenRedPen Yeeay!!!

[@blackpenredpen]

My pleasure!

[@jaywyn2584]

My textbook did a terrible job explaining this proof, and most proofs I found involve proving itself. Thanks for making this proof crystal clear. Well done.

[@reazraza]

Blackpenredpen white paper. You are uplifting the channel name. Good job!

[@blackpenredpen]

Yup!!!! Thanks!!!!

[@JM-hu3pk]

@@blackpenredpen love this vid. kinda irrelevant, but what pens do you use?

[@franseresplandor8591]

this is probably the best teacher I have in terms of all the trigonometic identities. Very simple and consice! Love it!

[@carterwoodson8818]

I am so happy you put this up 2 YEARS AGO!!! thank you so much, never investigated these relationships

[@crystalhuang8258]

I have been looking for a proof just like this and found it. Thank you so much!

[@humester]

5*: Everything I have found on the Internet so far has been a proof, not a derivation; meaning that they start with the answer and then simplify the answer on the RHS to equal the LHS. Yours is the first real derivation I've found. Love it!

[@diesitreinta]

This proof videos are my favorite, thanks! Love u ❤

[@gustavosedano294]

That demostrative videos are amazing!!!

[@blackpenredpen]

Thank you!!

[@BashirArsine]

After 11 months, still helpful!

[@BigDBrian]

an informal way to derive alpha in terms of A and B: Say for example we take sin(40) + sin(60). how would we determine alpha and beta? well, it's not too hard to see it'll be 50-10 and 50+10. alpha is fifty, because it's the average, and beta is 10, because its the difference between each term and the average.

[@julianarodrigues1861]

You make me so happy now, thanks a lot!

[@marcioamaral7511]

Pretty easy identity to prove but still a useful one

[@JJ-uj1wi]

Thank you! I have been find the proofs for this, since the precalculus lesson I attended didn't prove this for us

[@headmanlesetlhe1635]

beautiful magnificently explained.... thank you so much

[@sansamman4619]

THIS IS SO FUN! please keep on proving stuff and do more videos with this OG style!!

[@blackpenredpen]

San Samman OK!!!!!!

[@rithiek5446]

Wanted to say Thank you! Learnt a lot from you till date :)

[@BashirArsine]

Great video, thx.

[@florianvanbondoc3539]

That's just great! 😯

[@jagsingh2508]

Thank you very much!!! It becomes so much easier to memorise now that I know how it works, video very much appreciated!

[@copperfield42]

this is a very elegant proof. I on the other hand, started from the 2sin((a+b)/2)cos((a-b)/2) and arrive at sin(a)+sin(b): 2sin((a+b)/2)cos((a-b)/2) =2[cos(a/2)sin(b/2) + cos(b/2)sin(a/2)][cos(a/2)cos(-b/2) -sin(a/2)sin(-b/2) ] =2[cos(a/2)sin(b/2) + cos(b/2)sin(a/2)][cos(a/2)cos(b/2) +sin(a/2)sin(b/2) ] =2[ cos(a/2)sin(a/2)(cos^2(b/2)+sin^2(b/2)) + cos(b/2)sin(b/2)(cos^2(a/2)+sin^2(a/2)) ] =2[ cos(a/2)sin(a/2) + cos(b/2)sin(b/2) ] =2cos(a/2)sin(a/2) + 2cos(b/2)sin(b/2) =sin(2a/2) + sin(2b/2) =sin(a) + sin(b)

[@emmamjkang]

How did you get from =2[ cos(a/2)sin(a/2)(cos^2(b/2)+sin^2(b/2)) + cos(b/2)sin(b/2)(cos^2(a/2)+sin^2(a/2)) ] =2[ cos(a/2)sin(a/2) + cos(b/2)sin(b/2) ] ?

[@Lily-zd6dx]

thank you so much!!!

[@royler8848]

The sin (a+b) Vid isn't in the description

[@raiedahmednishat8883]

yah, that's what I looked for too

[@blackpenredpen]

Boypig24 sorry I forgot. It's here kzbin.info/www/bejne/aITPp36koburbbc

[@tariqhaiderbhuiyan2461]

Really helped me out!

[@baskard8018]

I like the video so much.

[@blackpenredpen]

Thank you!!!

[@alejrandom6592]

trigonometric proofs are beautiful

[@houwen7697]

this helped me understand! thank you

[@eleazaralmazan4089]

Can you make videos explaining how to solve equations involving the floor function? An example of such equation would be floor(x)-2floor(x/2) = 1. Great videos by the way!

[@tasninnewaz6790]

Everybody know this. your are my favourite teacher and i hoped that it will be a geometric explain.

[@blackpenredpen]

Thank you!!!

[@jimpal5119]

Where’s the dabbing man? Love the vids👌

[@BigDBrian]

no please

[@mandre2390]

bless ur soul

[@heisalima]

Thank you so much.

[@anything6889]

Cool job!!!

[@faridsalmanalfarisyi4099]

amazing

[@balyoz3349]

thank you so much

[@pchk1]

Very curious at 3:33 . . . Aww, why bother with the clumsy step of multiplying alpha - beta = B by negative one at all? Come on, simply SUBTRACT the whole thing from alpha + beta = A, and you IMMEDIATELY get 2beta = A - B Also, at 4:30 . . . No parentheses will be necessary for single-variable arguments in trigonometric functions, thus it is perfectly ok to write sinA + sinB rather than the, again very clumsy, sin(A) + sin(B) . . . especially that you were already writing in black and red ^_^ Finally, are you also on Facebook? I'd love to join you if you happen to be there!

[@wduandy]

Can you racionalize 1/[cuberoot(a)+cuberoot(b)+cuberoot(c)] please? Love your videos

[@MarkMcDaniel]

Did your school tell you that you can't use their classroom white board for videos any longer?

[@blackpenredpen]

Snarky Mark I live 42 miles away from my school.

[@MarkMcDaniel]

Ouch, quite the commute.

[@malathim2882]

But be clear with video clarity it's somehow blurrrr

[@aneeshsrinivas9088]

this identity makes it not circular reasoning to use lhopitals rule on lim_{h->0} (1-cos(h))/h, because you can derive the cosine versions of this identity by replacing a with a+π/2, and b with b+π/2, and for proving the derivitives of sine and cosine, just use these identities instead of expanding out via the sum and difference identities, d/dx(sin(x))=lim_{h->0} (sin(x+h)-sin(x))/h=lim_{h->0} (2cos((x+h+x)/2)sin((x+h-x)/2))/h=lim_{h->0} (2cos(x+h/2)sin(h/2))/h, and do the same thing for cosine

[@That_One_Guy...]

can you derive this formula using euler's formula ? (without subtituting alpha+beta = A and alpha-beta = B ?)

[@nikoladjordjevic2706]

Could you please do int_0^1 int_0^1 [1/(1-xy)] dx dy = zeta(2)? Thank you and of course great channel ;D

[@kobethebeefinmathworld953]

I wonder who thumbs down

[@ალექსანდრეოთხოზორია]

thanks brother proof of sum to product identities is not in my book for some reason

[@FJ-mn2pi]

谢谢

[@yenyelinito]

Link for those pens please lmao

[@jahanaraparveen9436]

e^xcosz-(1/3)e^(3x)cos(3z)+(1/5)e^(5x)cos(5z)-.... Please help me with this series.I am asked to find the infinite sum of this series.Got this from a complex variable book. :/

[@malathim2882]

First I was scared 😬💀 f d channel but it was very useful

[@smitashripad9757]

Does there exists something like that for cosine

[@blackpenredpen]

Yes! Use the sum and difference formulas for cosine and you can get the results.

[@kenichimori8533]

The point P ≒ P

[@dudenD8000]

老哥，听不懂啊

[@shrihari154]

For those who looking for video link mentioned in the video i.e formula for Sum of angles here it is: kzbin.info/www/bejne/aITPp36koburbbc

[@andrew4ig]

solve pls sin(3x)/cos(x)=39/41

[@indra9013]

超

[@hariskayani4703]

Can you solve Z^3 - 4j = 0

[@strangemathematician1572]

What did you ment by w and j? Are they random variables?if they are so you need at lest one more equation to solve them