View source for Exponential form

Every [[complex number]] <math>z</math> is the sum of a [[real]] and an [[imaginary]] component, <math>z=a+bi</math>.  If you consider complex numbers to be [[coordinate]]s in the [[complex plane]] with the <math>x</math>-axis consisting of real numbers and the <math>y</math>-axis [[pure imaginary number]]s, then any point <math>z=a+bi</math> can be plotted at the point as <math>(a,b)</math>.  We can convert <math>z</math> into [[polar form]] and re-write it as <math>z=r(\cos\theta+i\sin\theta)=r cis\theta</math>, where <math>r=|z| = \sqrt{a^2 + b^2}</math>.  By [[Euler's identity|Euler's formula]], which states that <math>e^{i\theta}=\cos\theta+i\sin\theta</math>, we can conveniently (yes, again!) rewrite <math>z</math> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number.

So <math>z=re^{i\theta}</math> looks like:
<asy>
import markers;
pair A,B,C,D,E;
A=(0,1);
B=(0,0);
C=(1,0);
D=(1/2,sqrt(3)/2);
E=(1/2,0);
draw(A--B--C);
label("Im",2*A/3,W);
label("Re",2*C/3,S);
dot("$z$",D,NE);
draw(B--D,blue);
draw(E--D,red);
draw(B--E,green);
label("$r$",(B+D)/2,p=blue,NW);
label("$\sin\theta$",(E+D)/2,p=red,E);
label("$\cos\theta$",(B+E)/2,p=green,S);
markangle("$\theta$",radius=13,C,B,D,ArcArrow,orange);
</asy>

==See also==
* [[Trigonometry]]
* [[Trigonometric identities]]

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[[Category:Complex numbers]]