# Thread: question about complex plane

1. ## question about complex plane

hi guys,

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

Could you please explain me? with one example please

2. Originally Posted by problady
I am wondering how come cos(theta)+i sin(theta) == x +yi
Could you please explain me? with one example please
Do you understand polar coordinates?
If you do, what is there to say about your question?
If not, then you should understand that this is not a tutorial service.

3. Yes I know about polar coordinates! So please explain...

4. Originally Posted by problady
Yes I know about polar coordinates! So please explain...
What is there to explain?
$\displaystyle \left( {x,y} \right) \equiv \left( {r\cos (\theta ),r\sin (\theta )} \right).$ Apply that to $\displaystyle x+\mathbf{i}y\equiv r\cos (\theta )+\mathbf{i}r\sin (\theta )$.

5. Thanks dear!

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 $\displaystyle r= \sqrt{x^2+ y^2}$ make a right triangle. The "opposite side", parallel to the y-axis, has length $\displaystyle y= r sin(\theta)$ and the "near side", along the x-axis, has length $\displaystyle x= rcos(\theta)$.

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 $\displaystyle x+ iy= r cos(\theta)+ i r sin(\theta)$. Of course, your example, $x+ iy= cos(\theta)+ i sin(\theta)$, implies that $r= |x+ iy|= 1$