1. ## question about complex plane

hi guys,

I am wondering how come cos(theta)+i sin(theta) == x +yi

I am wondering how come cos(theta)+i sin(theta) == x +yi
Do you understand polar coordinates?
If not, then you should understand that this is not a tutorial service.

$\left( {x,y} \right) \equiv \left( {r\cos (\theta ),r\sin (\theta )} \right).$ Apply that to $x+\mathbf{i}y\equiv r\cos (\theta )+\mathbf{i}r\sin (\theta )$.
6. Another way of looking at it: from the point (x, y) in the coordinate plane, draw a perpendicular to the x-axis. That, the x-axis itself, and the line from (x,y) to the origin, which has length $r= \sqrt{x^2+ y^2}$ make a right triangle. The "opposite side", parallel to the y-axis, has length $y= r sin(\theta)$ and the "near side", along the x-axis, has length $x= rcos(\theta)$.
The standard convention for representing complex numbers as points in the plane, sets the imaginary part of the number to the y-coordinate and the real part to the x-coordinate. That gives $x+ iy= r cos(\theta)+ i r sin(\theta)$. Of course, your example, $x+ iy= cos(\theta)+ i sin(\theta)$, implies that $r= |x+ iy|= 1$