Deriving the Infinite Product Representations for all the Trigonometric Functions!

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Sin Prod: • Deriving EULER's INFIN...
Good morning my dear children! Let us derive a lot of cool identities today! By starting off with the Cosine, we use the double angle formula to turn it into a quotient of sin waves. From there on out we can get ourselves the tangent, cotangent, etc.! Enjoy! =D
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Опубликовано:

4 июн 2019

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Комментарии : 64

@-aaa-aaa 5 лет назад

A video on the hyperbolic functions and maybe some complex-valued analogues would be greatly appreciated and I think interesting.

@AndrewDotsonvideos 5 лет назад

Instructions unclear. Have socks in my mouth and belt around my neck in a batman costume.

@Assault_Butter_Knife 5 лет назад

For cos(X) couldn't you just use the sin(X) expansion and then just substitute every X with (x-pi/2)?

@angelmendez-rivera351 5 лет назад

You could, but that would be super impractical.

@dexter2392 5 лет назад

yeah but that would be even messier and longer to write than using double angle identity for sine

@birupakhyaroychowdhury974 5 лет назад

Papa flammy is just excellent....👍🏼

@Gustavo_0107 5 лет назад

When will we have another engennering 101 pap?

@nicolassamanez6590 5 лет назад

pappa flammy, how come these products dont have a namesake like taylor or mclaurin series do? why arent they named after pappa wheeler?

@josiproak739 5 лет назад

Damn papa flammy, you better get serious ad revenue for this, youtube displayed 5 ads in the video :P

@ronanglemusic1993 5 лет назад

Hey Pappa, love the videos!

@ianmathwiz7 5 лет назад

If the sin infinity grill can be derived from the cotangent infinity boi, does that make the sin an infinity trans grill? Trigonometry: changing genders before it was cool.

@tanvec 5 лет назад

Have you ever slid into frame, then fallen square on your ass? Editor note: I watch for the math and laugh at the “bloopers”

@jackhanke343 5 лет назад

All the "drake" identities

@cameronkhanpour3002 5 лет назад

wouldn't it be easier to express sin(x) = x(product(cos(x/2^k))) with k=1 to infinity?

@hungryfareasternslav1823 5 лет назад

That would be infinite sum inside the infinite product!!!

@dectorey7233 5 лет назад

Papa Flammy making my slow 9-5's bearable :D

@PSNsomeonealive 5 лет назад

excellent video papa

@birupakhyaroychowdhury974 5 лет назад

No he's papa flammy....

@angelmendez-rivera351 5 лет назад

For tan(x), the answer is yes: you can move the limit to the front and the collect the terms for both product and cancel out the odd numbers, leaving only the even numbers, because once again, both products converge absolutely, and we know they do because you just proved them by derivation. Hence, yes, the tangent product is just the sine product but with even indeces instead of natural indeces.

@NintendoGamer789 2 года назад

Bruh what double check that, what you get is just 2sin(x/2). I don't think there is a nice simplification that exists for tanx.

@angelmendez-rivera351 2 года назад

@@NintendoGamer789 Yes, you are right, that was my mistake. I must have been thinking of something else. There is no cancellation of the terms. However, you can still bring the limit to the front. That part of my comment stays accurate.

@MrRyanroberson1 5 лет назад

secant is related to the secant line, therefore whichever trig function it is related to is just an accident. 1/cos happens to equal the length of a secant line

@tommasoseverini3767 5 лет назад

10:55 the proof is left as an exercise for the reader..... TRIVIAL!!!

@kaztarihtanu 5 лет назад

👍

@ericb7291 5 лет назад

Is the infinite product of (1-x^2/(pi^2*k^2)) equal to 1 for all x?

@soriac9898 5 лет назад

How about a video proving the riemann hypothesis ? :v

@MaxxTosh 5 лет назад

Hey Daddy Hotboi, if you can write out this infinite product for the trig functions, which are functions based on x^2 + y^2 = 1, and for the hyperbolic trig functions which are functions based on x^2 - y^2 =1, can you find infinity bois for all x^p +/- x^p = 1?

@sofianemohammed8048 5 лет назад

flammable maths Can you integrate tan x from 0 to π/2 😈 👿 ?

@sofianemohammed8048 5 лет назад

Flammable Maths yes i know but Dr πm integrated it with à hard method i want to see your method because my career ended so as to find the solution 😂😂 please do the video soon 😍

@rot6015 5 лет назад

@@sofianemohammed8048 😂😂😂😂😂😂😂😂😂😂😂😂😂😍😍😍😍😍😂😂😂😂😍😍😍😍😣😂😂😍😂😂😍😍😂😍😂😍

@sofianemohammed8048 5 лет назад

JJ - they evaluate diverges integrals like x^x and x^-x and 1/x from 0 to 1 etc ... so i hope that they evaluate the integral of tan x from 0 to π/2

@duncanw9901 5 лет назад

Probably doesnt get u anywhere but the denominator of the secant one is a difference of squares

@olli3686 2 года назад

sin(x)=cos((x-pi/2)(-1)^j+2*pi*k) = cos(x-pi/2). Couldn’t we substitute to get rid of the x by the infinite product?

@rot6015 5 лет назад

papa do you follow the season 3 of attack on titan?

@everlastingauraX 5 лет назад

2:50, use pepega, it sounds funnier. :D

@F-S. 3 года назад

Is there a product formula for the natural log?

@PapaFlammy69 3 года назад

yup! :)

@Noam_.Menashe 2 года назад

I think you can get it from the product of e^x. Not sure though.

@metecatl4586 2 года назад

very nice, but there is another way also.

@aweebthatlovesmath4220 2 года назад

Infinity boi

@mipmip4575 5 лет назад

what about the versine xD

@ianmathwiz7 5 лет назад

This proof is trivial and left as an exercise for the viewer :p

@tszhanglau5747 5 лет назад

You can say you have "speech 100" (I hope you get the reference)

@justkarl2922 2 года назад

You can tell infinity grills are not that much to be simplified

@orangeguy5463 5 лет назад

why isn't it simply cos(x) = sin(x + pi/2) =(x+pi/2)prod(1-(x+pi/2)^2/(kpi)^2) ? Or is this an equally valid, but not as useful product for cosine, when it comes to analytic number theory?

@angelmendez-rivera351 5 лет назад

It is equally valid, but as you may suspect, it is totally useless

@ai_serf 5 лет назад

Can someone please explain the engineer comic strip?

@leofisher1280 5 лет назад

Engineers often use sinx=x to estimate the values of sinx. It works pretty well for low values. (essentially its just the first value of the Taylor series).

@hamiltonianpathondodecahed5236 5 лет назад

*yes*

@phukaoprommolmard5282 5 лет назад

A S S U M E S M A L L A N G L E

@zackbartley3194 5 лет назад

i!??

@shandyverdyo7688 5 лет назад

Meh.. Infinity BOIZZZ Unch... unch... -_-

@koichi8529 5 лет назад

Are you German?

@koichi8529 5 лет назад

Flammable Maths Great. I think Deutschland is best country in the world.

@Mathmagician73 5 лет назад

Sir i have one beautiful, crazy question. I think you should make video on that question if you don't mind can i send u question. Pls give me your mail id or social media weblink . Sir trust me its really beautiful question. Thank you:)

@willnewman9783 5 лет назад

You say all of these things are part of analytic number theory, but you have yet to show us anything involving what I think of as number theory: integers, primes, algebraic numbers, etc.

@angelmendez-rivera351 5 лет назад

will newman Number theory and analytic number theory are not the same thing

@willnewman9783 5 лет назад

@@angelmendez-rivera351 From wikipedia: "In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers." What he has done does not prove anything about the integers.

@angelmendez-rivera351 5 лет назад

will newman Okay, and? Thank you for literally proving my one and only point: that they are not the same thing. It says "branch of number theory", not "the same as number theory." Stop trying all intellectual when you can't tell apart my claims from his and you're reading comprehension is so bad it proves my point yet you think it proves yours. I've said all that was needed to say. I'm done with this conversation. Bye.

@willnewman9783 5 лет назад

Let me translate that "Oh, I see that I am wrong, so I to pretend like I was write and end the conversation."

@aweebthatlovesmath4220 2 года назад

@@willnewman9783 theory that has numbers lol that's how i define it