# question about complex plane

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• Apr 3rd 2011, 08:43 AM
problady
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
• Apr 3rd 2011, 08:53 AM
Plato
Quote:

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.
• Apr 3rd 2011, 09:01 AM
problady
Yes I know about polar coordinates! So please explain...
• Apr 3rd 2011, 09:11 AM
Plato
Quote:

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 $\displaystyle x+\mathbf{i}y\equiv r\cos (\theta )+\mathbf{i}r\sin (\theta )$.
• Apr 3rd 2011, 09:15 AM
problady
Thanks dear!
• Apr 3rd 2011, 11:04 AM
HallsofIvy
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 $r= |x+ iy|= 1$