3DPolarForm

Writing a complex number as z = x + iy is referred to as ** Cartesian Form **.
 * Polar Form **

A different way of writing a complex number is ** Polar Form **.

In **__Polar Form__**, the position of the complex number is located on the Argand plane by stating [r, θ] where
 * r = distance from origin **{Modulus}**
 * **r is always positive**
 * θ = angle anticlockwise from positive real axis **{Argument}**
 * **{**q **undefined if r = 0}**

Modulus

math . \qquad r = mod(z) = |z| \\. \\ . \qquad r = \sqrt{x^2+y^2} math

{Also called the __**Absolute Value**__}

Your calculator uses the **|x|** function (on the MATH keyboard) to calculate the modulus.

Argument

math . \qquad \theta = \text {Arg}(z) \\. \\ . \qquad \theta = Tan^{-1} \left( \dfrac{y}{x} \right) \qquad \theta \in \left( -\pi, \pi \right] math

Note: The ** Principal Domain ** for Arg(z) is (– p, p ]

Whenever you provide an argument or a complex number in polar form, you __must__ make sure the angle is within the Principal Domain.

Your calculator has an **arg** function in the ACTION menu, COMPLEX submenu

CIS Notation

Using basic trig, we get

math \\ . \qquad x = r \, \cos \big( \theta \big) \\. \\ . \qquad y = r \, \sin \big( \theta \big) math

so math . \qquad z = x + yi math

becomes math . \qquad z = r \, \cos \big( \theta \big) + r \, i \, \sin \big( \theta \big) \\. math math . \qquad z = r \Big( \cos \big( \theta \big) + i \, \sin \big( \theta \big) \Big) math

**Abbreviation:** math . \qquad z = r \, \text{cis} \big( \theta \big) math


 * Example 1a: **

Write the following complex number in polar form :

math . \qquad z = 4 + 4 \sqrt{3} \, i math

First find r and θ math \\ . \qquad r = \sqrt{4^2+(4\sqrt{3})^2} \\. \\ . \qquad \quad = \sqrt{16+48} \\. \\ . \qquad \quad =\sqrt{64} \\. \\ . \qquad \quad =8 math

math \\ . \qquad \theta = Tan^{-1} \left( \dfrac {4\sqrt{3}}{4} \right) \\. \\ . \qquad \quad = Tan^{-1} \left( \sqrt{3} \right) \\. \\ . \qquad \quad = \dfrac {\pi}{3} \\. \\ math

Complex number is in 1st Quadrant so no need to change angle.
 * Remember Principal Domain is (– p, p ]

Now write in __**cis**__ form

math . \qquad z = 8 \, \text {cis} \Bigg( \dfrac {\pi}{3} \Bigg) math

Your calculator can convert from Cartesian form to Polar Form using the __**compToTrig**__ command which is in the ACTION menu, COMPLEX submenu.

NOTE:
 * The ** complex conjugate ** of z is obtained by making the argument negative

math . \qquad \bar z = 8 \, \text {cis} \Big( -\dfrac {\pi}{3} \Big) math


 * Example 1b: **

Write the following complex number in polar form:

math . \qquad -2 + 2\sqrt{3} \, i math


 * Solution:**

First find r and θ

math \\ . \qquad r = \sqrt{ \Big( -2 \Big)^2 + \Big( 2\sqrt{3} \Big)^2 } \\. \\ . \qquad \quad = \sqrt {4 + 12} \\. \\ . \qquad \quad = \sqrt{16} \\. \\ . \qquad \quad = 4 math

math \\ . \qquad \theta = \tan^{-1} \Bigg( \dfrac{2\sqrt{3}}{-2} \Bigg) \\. \\ . \qquad \quad = \tan^{-1} \Big( -\sqrt{3} \Big) \\. \\ . \qquad \quad = -\dfrac{\pi}{3} \qquad \textit{ according to the calculator} math

math \\ \textbf{BUT } -2 + 2\sqrt{3} \, i \; \textit{ is in the 2nd quadrant} \\ \\ \textit {and } -\dfrac{\pi}{3} \; \textit{ is in the 4th quadrant} math

SO {adjust θ to be in the 2nd quadrant}

math . \qquad \theta = \pi - \dfrac{\pi}{3} = \dfrac{2\pi}{3} math

Now write in cis form

math . \qquad z = 4 \, \text{cis} \Bigg( \dfrac{2\pi}{3} \Bigg) math

Changing From Polar Form back to Cartesian Form

Given that z = r cis(q )
 * write this as z = r ( cos(q ) + i sin(q ) )
 * then use exact values or calculator to find the values

NOTE: Take care to use +/– correctly depending on the quadrant q is in.

Your calculator can convert from Polar Form back to Cartesian form using the __**cExpand**__ command in the ACTION menu, COMPLEX submenu.


 * Example 2 **

Convert into cartesian form: math . \qquad z=2 \, \text{ cis } \Big( \dfrac{5\pi}{6} \Big) math


 * Solution:**

math . \qquad z = 2 \, \Big( \cos{\dfrac{5\pi}{6}} + i \, \sin{\dfrac{5\pi}{6}} \Big) math

{Now use exact values. Angle is in 2nd Quadrant so sin is +ve and cos is –ve}

math \\ . \qquad z = 2 \, \Big( - \dfrac{\sqrt{3}}{2} + \dfrac{1}{2} \, i \Big) \\. \\ . \qquad z = - \sqrt{3} + i math

Alternate Polar Form (not in course but useful)

z = rcis(θ) can also be written as z = re θi .. {θ in radians}

This is the form used on many calculators.

To change a complex number to polar form, use the __**compToTrig**__ command in the ACTION menu, COMPLEX submenu. (see screen to the right)

To see the alternate form of a polar number, use the __**compToPol**__ function in the ACTION menu, COMPLEX submenu.

On your calculator math . \qquad \text {compToPol} \, ( 4 + 4\sqrt{3} \, i ) math

should give the result: math . \qquad 8 \, e^{ \dfrac {\pi i}{3} } math

Properties of the Modulus

Given math . \qquad |z| = \sqrt{x^2+y^2} math

square both sides math \\ . \qquad |z|^2 = x^2+y^2 \\. \\ . \qquad \quad= (x+yi)(x-yi) \\. \\ . \qquad \quad = z \,\bar{z} math

This is a useful rule: math . \qquad |z|^2 = z \, \bar{z} math

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