5 ms·
My favorite is a way to prove the sum of angles formula for sine and cosine. cos(x1+x2) + isin(x1+x2) = e^((x1+x2)i) = e^(i*x1) *
by malisper 13y ago
My favorite is a way to prove the sum of angles formula for sine and cosine.
cos(x1+x2) + isin(x1+x2) = e^((x1+x2)i)
= e^(i*x1) * e^(i*x2)
= (cos(x1) + isin(x1)) * (cos(x2) + isin(x2))
= cos(x1)cos(x2) - sin(x1)sin(x2)
+ i*(sin(x1)cos(x2) + sin(x2)cos(x1))
Since the imaginary parts have to be equal and the real parts have to be equal:
sin(x1+x2) = sin(x1)cos(x2) + sin(x2)cos(x1)
cos(x1+x2) = cos(x1)cos(x2) - sin(x1)sin(x2)
- NAFV_P 13y agoYeah, that's a more generic form of what I said. ok, hold on... e^(i*x)= cos(x)+i*sin(x) e^(-i*x)= cos(-x)+i*sin(-x)= cos(x)-i*sin(x) cos(x)= (1/2) * (e^(i*x)+e^(-i*x)) sin(x)= (-i/2) * (e^(i*x)-e^(-i*x)) cos(i*x)= (1/2) * (e^(i*i*x)+e^(-i*i*x))=(1/2) * (e^x+e^(-x)) sin(i*x)= (-i/2) * (e^(x)-e^(-x)) That's the amusing part.