>>11600108It does not rely on handwaving and is not more complicated than a diagram. It seems like you are a bit too low-IQ to get the idea of the proof just by reading anon's post. Here, let me elucidate it.
1. You realize that cosa, sina are the components of the point in the plane that's rotated by angle a. This is the unit circle definition of sin, cos. Absolutely nothing new.
2. You can add points on the plane by the formula (a,b)+ (c,d)=(a+b,c+d). It's a simple formula and it corresponds to drawing a parallelogram with sides 0-(a,b), 0-(c,d) and looking where the end of the diagonal lands. Again, this is very elementary and doesn't take more than 1 minute to explain.
3. Rotations are nice. They map parallelograms to parallelograms and squares to squares.
4. But that means that if you have a rotation R, then R(x,y)= R(x,0) + R(0,y) since (x,y)=(x,0)+(0,y). This is all about decomposition: it makes rotations much easier to calculate since you only need to rotate points lying on the x and y axes!
5. The point (cos(a+b), sin(a+b)) is the result of rotating the point (cos(a), sin(a)) by angle b. Let R be the rotation by angle b. Then R(1,0)=(cosb, sinb) and R(0,1)=(-sinb, cosb).
Hence we calculate: (cos(a+b), sin(a+b)) = R(cosa, sina)= R(cosa, 0) + R(0, sina) = cos(a) (cos(b), sin(b)) + sin(a) ( -sin(b) , cos(b))=(cosacosb - sinasinb, cosa sinb + sina cosb).
As you can see, the result pops out itself!
Easy, intuitive, rigorous, elegant. The only idea you need to remember is : "Rotations linear, decompose".
Where do you see the handwaving?
>>11600674Exactly! That's why I posed the challenge. Both the complex analysis proof and the angle chase proof are disgusting and doesn't get to the meat of the issue: rotations of the plane are nice!
>>11600152Absolutely every fifth grader knows what rotations are and vectors don't take longer than a minute to explain. The complex analysis proof, however, relies heavily on stuff that will go WAY over the fifth-grader's head.
