This Changes Everything
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- เผยแพร่เมื่อ 10 ธ.ค. 2024
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AC 2024 Playlist: • This Changes Everything
Whale come to the first episode of the advent calendar :) Today we derive the addition formula for the tangent, one of my most favourite trig identities. Enjoy! =D
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Cosine of wah
Me needing a bottle of wah right nah
wah equils ecks squaed
Cosine Siwa
Nothing like watching 7 and a half minutes of flammy guy writing math on blackboard that grows slowly more and more incomprehensable at 2 in the morning
I love how many trig idenities can be recovered from e^(ix) = cis(x)
Such a beautiful identity.
Though you need the angle-sum-identity of sine and cosine to proof Euler's formula
@@sternli728 Not necessarily.
Premise 1: d(e^ix)/dx = i e^ix (By chain rule)
Premise 2: d(cos(x))/dx = -sin(x)
Premise 3: d(sin(x))/dx = cos(x)
Premise 4: d(f(x) + g(x))/dx = df(x)/dx + dg(x)/dx
Premise 5: If there exist constants a, b and k such that f'(x) = ax + bf(x), g'(x) = ax + bg(x), and f(k) = g(k), then f(x) = g(x) for all x.
Define f(x) = e^ix. Note that f'(x) = i e^ix = i f(x). Also note that f(0) = 1.
Define g(x) = cos(x) + i sin(x). Note that g'(x) = -sin(x) + i cos(x) = i(i sin(x) + cos(x)) = i g(x). Also note that g(0) = 1.
So, f'(x) = 0x + if(x), g'(x) = 0x + ig(x) and f(0) = g(0) = 1. From premise 5, this is adequate to show that f(x) = g(x). Thus e^ix = cos(x) + i sin(x).
@chaosredefined3834 But how do you prove Premise 2 and 3 without using the angle sum identity?
@@sternli728 Another proof is using the Maclaurin series. Using the expansion for the exponential function, sub x=ix, and split the resulting series into two separate real and imaginary series. You will find the real part is the Maclaurin series for the cosine function and the imaginary part is the series expansion for the sine.
Love this, De Moivre’s Theorem is underrated
Always nice to start my day with some clean, simple trigonometry on one hand and an open book on string theory in the other 😂
Holy sh*t! The Advent calendar is beckk!!
Cool, the advent calendar is back! Please do another meme review with Andrew Dotson.
Very nice. Love how 'Daddy Euler' makes it all super easy 🙈
I missed Papa Flammys Advent Calender. Best Time of the year
My favorite identity is the one of arctan(x) + arctan(y) = arctan(x+y/1-xy) because of the relationship between this other formula: tan(x+y) = tan(x) + tan(y)/1 - tan(x)tan(y) (They look pretty similar)
You're too late my friend! It's already the 2nd here in Australia!
Have a Foster’s for all of us!!
5:55 UNCLE JIMBO
Hell Ned, everything's legal in Mèxico!
Hell yea, advent calendar time!
Isn't the derivation at 1:39 circular? The Taylor series and thus the derivatives of sine and cosine are needed to prove Euler's formula, but the angle sum formulas are needed to find the derivatives of sine and cosine.
I think the point was to present a quick way to remember the identities if you forget them rather than proofs of the identities. Seen as the proofs using constructions out if triangles take at least ten minutes.
My favorite part of the year is papa flammys advent calendar
Let’s gooooo it’s back!
Omg bae wake up Advent Calendar is here!
You clickbaited me grrrrrrrrr.
is it worth to go for a math phd?? im currently in 11th grade and already learnt like the first 3 semesters of the study, and its fun
Don't ask if its worth it. If you want to, do it. Besides, if you are in grade 11, you still got all of university to figure that out.
@@david4649 Fair enough. I guess I'm just slightly overwhelmed by possibilities
I had similar questions before going to college. I disagree with David's comment in that I think it's really important to consider if it's worth it, and also like David said, you'll have a lot of time to decide if you want to go down that route. In college you get more freedom with classes you choose and there's a lot more freedom in general and at least for me, being able to explore naturally resolved all of my concerns. You have really limited information right now about if you'll like college and/or benefit from it. If I recall correctly, on average, people switch college majors multiple times, so it's really normal to not lock in early (though if you're able to lock in early that can be good but most folks just don't know up front). People develop and change a lot in college and what you want in 3-6 years may be very different from what you want now. Just my thoughts. It sounds like you are ready to take college seriously though and you seem like a proactive person so I think you'll do awesome in any case. You got this! Sorry for not directly addressing your question, I think the unfortunate (or fortunate?) answer though is that you'll get more information along the way through exploring and along the way considering if it's worth it for you.
As a person who tends to overthink/overcalculate, hearing a really good professor say "You don't have to figure out your whole life right now; the rest of your life is for future you to decide" was one of the most valuable pieces of advice for me personally
@@projectpiano5231 thanks for your words. The quote at the end really is something to be considered
I thought everybody knew this
It's baaa-aaaack!
Pronunciation of y is driving me crazy
me too. Why does he do that?
Nice 👍
I'm early too
Omg hi
this was clickbait
holy shit i’m early