Deriving the Infinite Product Representations for all the Trigonometric Functions!

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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

[-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.

[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)?

[Gustavo_]

When will we have another engennering 101 pap?

[kummer]

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.

[nicolassamanez]

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

[ianmathwiz]

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.

[josiproak]

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

[cameronkhanpour]

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

[angelmendez-rivera]

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.

[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”

[MrRyanroberson]

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

[dectorey]

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

[tommasoseverini]

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

[F-S.]

Is there a product formula for the natural log?

[soriac]

How about a video proving the riemann hypothesis ? :v

[ericb]

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

[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?

[olli]

= cos(x-pi/2). Couldn’t we substitute to get rid of the x by the infinite product?

[sofianemohammed]

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