**11-2 Geometric and Trigonometric
Representation of Complex Numbers**
Imaginary axis
real
axis
*The above diagram is an Argand diagram. Notice
that the real numbers are on the x-axis and the imaginary numbers are on
the y-axis. Finding imaginary numbers in this plane are as easy as
finding points in the real plane. In the form a + bi, a is the real
part and b is the imaginary part. Move a units right or left ( depending
on + or - ) and b units up or down*

*(depending on + or -).*

*Rectangular form: z = a +
bi*

*Polar form: z = r cos *~~0~~
+ (r sin ~~0~~)i (remember a = r cos ~~0~~ and
b = r sin ~~0~~ substitute these in and presto!)

*factor out the r and get:
z = r(cos *~~0~~ + i sin ~~0~~)

*Math shorthand looks like
this: z = r cis *~~0~~

*The absolute value of z (the
distance from any point to the origin) =*

*
_______*

*| z | = \/ a*^{2}
+ b^{2} (good
old Pythagorus again)

__Example problems:__
**1) Express 3 cis 50**^{o}
in rectangular form.

** **__Solution:__
using the fact that a = r cos ~~0~~ and b = r sin ~~0~~
implies that

**
a = 3 cos 50 and b= 3 sin 50. Using your calculator gets us**

**
a = 1.93 and b = 2.30 with both answers rounded to hundredths.**

**
**__Therefore the answer is: 1.93 + 2.30i__

**2) Express -1 -2i in polar
form.**

**
__________ __**

** **__Solution:__
Use the fact that r = \/(-1)^{2}
+ (-2)^{2} r = \/ 5
= 2.24. Now use the fact that the Tan ~~0~~ = (y/x)
(see page 1 if you forgot!!). ~~0~~ = tan^{-1}(2/1).
Using your calculator gives us 63.4^{o}. Add 180 (why? we
are in the 3rd quadrant!!!) 63.4 + 180 = 243.4

**
Therefore, the answer is: 2.24 cis 243.4**^{o}

*To Multiply two complex numbers
in polar form:*

*1) Multiply their absolute
values*

*2) Add their polar angles.*

*In math terms if z*_{1}
= r cis w and z_{2} =
s cis y then z_{1}z_{2}
= rs cis ( w + y)

__Example problem:__
*1) Express (5cis 30*^{o})(7cis60^{o})
in polar and rectangular form.

* *__Solution:__
Polar form first: multiply the radii and add the angles.

*
*__Answer in polar form: 35 cis 90__^{o}

*
Now change this to get the rectangular form.*

*Remember, a = r cos *~~0~~
and b = r sin ~~0~~

*so a = 35 cos 90 = 0 and
b = 35 sin 90 = 35.*

__Answer in rectangular
form is: 0 + 35i__

**Next
section involves finding the powers of complex numbers!!**