Angle Sum Formula Proof with Linear Algebra!

Here we prove the angle sum formulas for sine and cosine. We can calculate sin(a+b) and cos(a+b) using the power of the rotation matrix, which is a really easy way to remember the equations!

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[PM-]

Flammable Maths proves the rotation matrix with the angle sum formula, this channel proves the angle sum formula with the rotation matrix.

[JSSTyger]

Euler's formula can also be used. e^[i(a+b)] = cos(a+b)+isin(a+b). Also, e^[i(a+b)] = e^(ia)e^(ib) = = The real parts = cos(a+b). The imaginary parts = sin(a+b).

[losiu]

Great video! It's even simplier when you use complex numbers. Complex number is another representation of an arrow (vector in 2d), multiplying complex numbers means rotation of the one arrow by the angle of the second (and scaling). So instead of arrow <cos(alpha), sin(alpha)> you have cos(alpha) + i*sin(alpha). And instead of multiplying vector times matrix, you multiply you complex number by e^(i*beta) (our rotor) and it's equal cos(beta) + i*sin(beta). ( cos(alpha) + i*sin(alpha) ) * ( cos(beta) + i*sin(beta) ) . After multiplication you get the same formulas. Beauty of maths - the same thing represented in 2 ways.

[FromTheMountain]

I hate having to remember trig identities, so I'll definitely be using this in the future! By the way, that paranthesis appearing at 1:53 scared me at first.

[BuddyNovinski]

Not only does this proof explain the angle sum formula, but it also emphasizes what a matrix does to vectors in the first place. I'm having flashbacks to the mid-1970's and wondering why both were never so presented this way.

[nuklearboysymbiote]

Flexing that perfect freehand quarter circle👌

[MathManMcGreal]

That's a fun connection between topics!

also, I saw the ) at the end! 😂

[adambascal]

Nice! I prefer this proof to the complex number one which just assumes Euler's identity :)

[kingbeauregard]

Waaaait a second ... where does that rotation matrix come from? Because I always assumed it was a convenient way of representing the angle sum formulas.

Also, it's not important, but the beard worked for you. You look good clean-shaven as well, but Beardy Mu was solid.

[pinklady]

Lovely maths!!! I wish there were more maths like this: challenging, interesting, yet simple. 😍😍😍 This week, I am revising trignometry and linear algebra this week.
Big thank you for this tutorial.

[tomkerruish]

Nice! Personally, I prefer using Euler's formula, but then I'm partial to analysis. (Have you seen the Epic Math Time video "If Math Subjects were People"? Trust me, you'll get a kick out of it.)
Did you make sure to play "The Ride of the Valkyries" at 7am every morning of finals week? Here's to hoping you can attend in person next year!

[violintegral]

I just derive the angle sum formulas from the double angle formulas