Converting Polar Complex Number to Rectangular Complex Numbers by xri35382

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									Trigonometry
Converting Rectangular & Polar Forms of Complex Numbers

A complex number has a Real component and an imaginary component.
In the rectangular coordinate system, a complex number is in the form a + bi .
In the polar coordinate system, a complex number is in the form r ( cos θ + i sin θ ) , or rcisθ .

Converting Polar Complex Number to Rectangular Complex Numbers
We convert polar complex numbers to rectangular form the same way that we convert polar
points to rectangular points:
                                         a = r cos θ
                                                 b = r sin θ

                         ⎛     4π         4π     ⎞           4π
Example:       Convert 2 ⎜ cos    + i sin        ⎟ , or 2cis    , to rectangular form.
                         ⎝      3          3     ⎠            3

                         4π     ⎛ 1⎞
                a = 2 cos   = 2 ⎜ − ⎟ = −1
                          3     ⎝ 2⎠
                         4π    ⎛    3⎞
                b = 2sin       ⎜ 2 ⎟=− 3
                            = 2⎜ −    ⎟
                          3    ⎝      ⎠

Converting Rectangular Points to Polar Points
We convert polar complex numbers to rectangular form the same way that we convert polar
points to rectangular points:
                                              r = a 2 + b2
                                                       ⎛b⎞
                                              θ = tan −1 ⎜ ⎟ **
                                                       ⎝ ⎠a

** Principal values for inverse tangent are in quadrants 1 and 4, plot (a, b) as an ordered pair,
determine the quadrant and adjust the answer for θ accordingly.

Example:       Convert −4 + 2i to polar form.

                r=    ( −4 )       + 22 = 20 = 2 5
                               2



                          ⎛ 2 ⎞
                θ = tan −1 ⎜   ⎟ = −26.6°
                          ⎝ −4 ⎠
               (-4, 2) is in quadrant 2, the angle with a reference angle of 26.6 is 153.4

                2 5 ( cos153.4 + i sin153.4 ) or 2 5cis153.4

								
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