Deriving the Infinite Product Representations for all the Trigonometric Functions!

[@-aaa-aaa]

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

[@AndrewDotsonvideos]

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

[@cameronspalding9792]

What the heck?!

[@Assault_Butter_Knife]

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

[@angelmendez-rivera351]

You could, but that would be super impractical.

[@dexter2392]

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

[@Gustavo_0107]

When will we have another engennering 101 pap?

[@kummer45]

This is a very delicate subject that requires Tannery’s theorem limit swap for infinite series and infinite products. It’s an advanced topic in complex analysis, real analysis and the study of series and products. It’s good to have an intuition like this.

[@birupakhyaroychowdhury974]

Papa flammy is just excellent....👍🏼

[@nicolassamanez6590]

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

[@cameronkhanpour3002]

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

[@hungryfareasternslav1823]

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

[@tanvec]

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

[@josiproak739]

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

[@ianmathwiz7]

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.

[@angelmendez-rivera351]

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]

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]

@@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.

[@RM-gc8lx]

@@angelmendez-rivera351 Then delete your comment, or edit it, you muppet.

[@MrRyanroberson1]

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

[@ronanglemusic1993]

Hey Pappa, love the videos!

[@tommasoseverini3767]

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

[@dectorey7233]

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

[@ericb7291]

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

[@F-S.]

Is there a product formula for the natural log?

[@PapaFlammy69]

yup! :)

[@Noam_.Menashe]

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

[@olli3686]

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?

[@jackhanke343]

All the "drake" identities

[@PSNsomeonealive]

excellent video papa

[@birupakhyaroychowdhury974]

No he's papa flammy....

[@MaxxTosh]

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?

[@orangeguy5463]

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]

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

[@rot6015]

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

[@sofianemohammed8048]

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

[@sofianemohammed8048]

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]

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

[@sofianemohammed8048]

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

[@ai_serf]

Can someone please explain the engineer comic strip?

[@leofisher1280]

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]

*yes*

[@soriac9898]

How about a video proving the riemann hypothesis ? :v

[@duncanw9901]

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

[@QasqaZhol]

👍

[@everlastingauraX]

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

[@justkarl2922]

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

[@tszhanglau5747]

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

[@metecatl4586]

very nice, but there is another way also.

[@aweebthatlovesmath4220]

Infinity boi

[@phukaoprommolmard5282]

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

[@shandyverdyo7688]

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

[@zackbartley3194]

i!??

[@Mathmagician73]

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:)

[@koichi8529]

Are you German?

[@koichi8529]

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

[@willnewman9783]

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]

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

[@willnewman9783]

@@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]

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]

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

[@aweebthatlovesmath4220]

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