Addition Formulae using Matrices?! - Complex Numbers, but Different [ Bonus 1 ]

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Todaay my good friend Sam from Whatthehectogon wants to show youi the art of deriving trigonometric addition formulas for the sin and cosine, using complex numebrs in matrix form! =) Don't forget subscribing to his channel :)

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[Nickesponja]

Now I don't know who is hotter, papa Flammy or papa Sammy

[benjaminbrady]

I mean. The angle sum tangent doesn't normally have an insanely pretty form and you just divide the angle sum sin and cosine forms. So just multiply sin matrix with inverse cosine matrix

[RCSmiths]

Matrices are one of my favorite portions of mathematics. Such a coincidence I firmly believe. Great discussion and work here as always, and amazing to see Sam here

[gergodenes]

Friendly reminder that you can define numbers as 1x1 matrices as well.
My question is: How does complex analysis work when we're using this kind of representation of complex numbers? How do we take complex derivatives and make line integrals with this way of dealing with complex numbers?

[konradswart]

One of the things that distinguishes a great mathematician to one that is not so great, is _testing the result after yo have gotten it!_


Therefore, I do not agree with dismissing the second column. At least it is there for testing.


That is not all. Thanks to the presence of the second column, we can see that complex numbers are not numbers at all. They are also not vectors. No, they are _operators on vectors!_ If you leave this second column out, you are eliminating this insight.


Apart from that, a good video!

[NuncNuncNuncNunc]

Love the videos that are orthogonal to the norm.

[liyi-hua]

e^Iα = 1*cos(α) + I*sin (α) is a wonderful discovery in 2 by 2 matrix. But the rest of the work seems trivial if you treat the result as a rotation matrix.
Thanks for sharing though.

[zaraal-ghnai]

Wth "What the Hectogon!"
Wtf "What the Function!"

[JeffreyMarshallMilne]

Frick ya! Luv hectogon.. luv papa flammy.. couldn't have asked for anything more

[JaybeePenaflor]

First reason why I watched--I wanted to know what happened to Papa Flammy.

[alkashi]

actually I’m looking for a bag and the only one (the « Trivial » one ) has a word which have a bad interpretation in my language so I’m here to ask if you could make an other bag like with infinity boy, it would be perfect :D
Thanks for reading and I love your channel 💘

[tszhanglau]

Papa looks different... Wait, it's what the hectogon

[mskiptr]

This video was added twice to the playlist!

[nikhilnagaria]

9:46 wth is that way of writing exp tho

[luisramrod]

Wow I shouldn't have ate that mushroom 😂

[MrRyanroberson]

Show: tan(a+b)(1-tan(a)tan(b)) = tan(a)+tan(b)

i tan(a+b)(1-tan(a)tan(b))
= (e^i(a+b)/cos(a+b) - 1) * (1 + (e^ia/cos(a) - 1)(e^ib/cos(b) - 1))
= (e^i(a+b)/cos(a+b) - 1) * (2 -e^ia/cos(a) -e^ib/cos(b) +e^i(a+b)/cos(a)cos(b))
= 2e^i(a+b)/cos(a+b) -e^i(2a+b)/cos(a+b)cos(a) -e^i(a+2b)/cos(a+b)cos(b) -2 +e^ia/cos(a) +e^ib/cos(b) -e^i(a+b)/cos(a)cos(b)
= i tan(a) + i tan(b) + e^i(a+b)/cos(a+b) * (2 - e^ia/cos(a) - e^ib/cos(b) + e^i(a+b)/cos(a)cos(b) - cos(a+b)/cos(a)cos(b))

If e^i(a+b)/cos(a+b) * (2 - e^ia/cos(a) - e^ib/cos(b) + e^i(a+b)/cos(a)cos(b) - cos(a+b)/cos(a)cos(b)) = 0, then tan(a+b)(1-tan(a)tan(b)) = tan(a)+tan(b).
Show: e^i(a+b)/cos(a+b) * (2 - e^ia/cos(a) - e^ib/cos(b) + e^i(a+b)/cos(a)cos(b) - cos(a+b)/cos(a)cos(b)) = 0

e^i(a+b)/cos(a+b) * (2 - e^ia/cos(a) - e^ib/cos(b) + e^i(a+b)/cos(a)cos(b) - cos(a+b)/cos(a)cos(b)) = 0, e^i(a+b) never 0, divide by it.
sec(a+b) * (2 - e^ia sec(a) - e^ib sec(b) + e^i(a+b) sec(a)sec(b) - cos(a+b)sec(a)sec(b)) = 0.
sec(a+b) * ((1 - e^ia sec(a))(1 - e^ib sec(b)) + 1 - cos(a+b)sec(a)sec(b)) = 0. assume cos(a+b) not 0, multiply by it.
((1 - e^ia/cos(a))(1 - e^ib/cos(b)) + sin(a)sin(b)/cos(a)cos(b)) = 0 = (i tan(a) i tan(b) + tan(a)tan(b)) = -tan(a)tan(b)+tan(a)tan(b)

[michaelschneider]

I got an ad for Dollar shave club...and it was a woman...trying to sell me a razor...I'm not a woman. Stupid youtube