The polar form

Complex numbers, grade 12 (5 units) Lesson 9 of 18Learning

The polar form

The modulus and the angle

The point of $\displaystyle {z}={a+bi}$ in the Gauss plane can also be described by its distance from the origin, $\displaystyle {r}={|z|}$ , and by the angle $\displaystyle \theta$ that its vector makes with the positive direction of the real axis (anticlockwise). In the right-angled triangle whose hypotenuse is $\displaystyle r$ :

$\displaystyle {a}={r\cos\theta}$ and $\displaystyle {b}={r\sin\theta}$ .
So $\displaystyle {z}={r\cos\theta+ir\sin\theta}$ , and this is the polar form (the trigonometric form) of $\displaystyle z$ :

\displaystyle z=r(\cos\theta+i\sin\theta)

\displaystyle r=|z|>0

In short it is written $\displaystyle r\operatorname{cis}\theta$ – from cos, i, sin.
The angle $\displaystyle \theta$ is called the argument of $\displaystyle z$ , $\displaystyle \arg z$ . It is fixed up to a multiple of $\displaystyle 2\pi$ : $\displaystyle {\operatorname{cis}\frac{\pi}{3}}={\operatorname{cis}\frac{7\pi}{3}}$ . Usually $\displaystyle 0\le\theta<2\pi$ is chosen.
Zero has no angle, so it has no polar form.

The angles are in radians, as on the formula sheet. In the expression cards $\displaystyle \operatorname{cis}\frac{\pi}{3}$ is read in radians, and $\displaystyle \operatorname{cis}60^{\circ}$ – with the degree sign – in degrees.

From the algebraic form to the polar

$\displaystyle {r}={\sqrt{a^2+b^2}}$ , and $\displaystyle {\tan\theta}={\frac{b}{a}}$ . $\displaystyle \tan\theta$ has two angles between $\displaystyle 0$ and $\displaystyle 2\pi$ , so choose by the quadrant of the point. The example below writes $\displaystyle {z}={-1+\sqrt{3}i}$ in polar form:

$\displaystyle {r}={\sqrt{1+3}}={2}$ .
$\displaystyle {\tan\theta}={\frac{\sqrt{3}}{-1}}={-\sqrt{3}}$ : $\displaystyle {\theta}={\frac{2\pi}{3}}$ or $\displaystyle {\theta}={\frac{5\pi}{3}}$ .
The point $\displaystyle (-1,\sqrt{3})$ is in the second quadrant, so $\displaystyle {\theta}={\frac{2\pi}{3}}$ .
$\displaystyle {z}={2\operatorname{cis}\frac{2\pi}{3}}$ .

Numbers on the axes, without calculating: $\displaystyle {1}={\operatorname{cis}0}$ , $\displaystyle {i}={\operatorname{cis}\frac{\pi}{2}}$ , $\displaystyle {-1}={\operatorname{cis}\pi}$ , $\displaystyle {-i}={\operatorname{cis}\frac{3\pi}{2}}$ .

From the polar form to the algebraic

Put in the values of $\displaystyle \cos\theta$ and $\displaystyle \sin\theta$ and open the brackets:

$\displaystyle {4\operatorname{cis}\frac{5\pi}{6}}={4\left(-\frac{\sqrt{3}}{2}+\frac{1}{2}i\right)}={-2\sqrt{3}+2i}$ .

For an angle with no exact value, round, and write “≈” instead of “=”. In the example below $\displaystyle {z}={3+4i}$ , in the first quadrant:

$\displaystyle {r}={5}$ , and $\displaystyle {\theta}={\tan^{-1}\frac{4}{3}\approx0.927}$ .
$\displaystyle z\approx5\operatorname{cis}0.927$ .

Careful: the shape of the polar form

In the polar form $\displaystyle r$ is positive, and $\displaystyle \cos$ and $\displaystyle \sin$ have the same angle, with $\displaystyle +$ between them:

$\displaystyle -2\operatorname{cis}\frac{\pi}{3}$ is not in polar form. $\displaystyle {-1}={\operatorname{cis}\pi}$ , so it is $\displaystyle 2\operatorname{cis}\frac{4\pi}{3}$ : a further half turn.
$\displaystyle {\cos\theta-i\sin\theta}={\cos(-\theta)+i\sin(-\theta)}={\operatorname{cis}(-\theta)}$ , since $\displaystyle \cos$ is even and $\displaystyle \sin$ is odd.

Exercise 1

Write in polar form. The angle in radians.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 2

Write in algebraic form.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 3

Write in polar form: $\displaystyle r$ positive, and the same angle in $\displaystyle \cos$ and $\displaystyle \sin$ with $\displaystyle +$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Exercise 4

Write in polar form approximately: round the angle to three decimal places, and write “≈”.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

Formulas: complex numbers, grade 12 (5 units)

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The algebraic form and the four operations

The number $\displaystyle i$
$\displaystyle i^{2}=-1$
The algebraic form
$\displaystyle z=a+bi$ $\displaystyle a=\operatorname{Re}(z),\ b=\operatorname{Im}(z)$
Equality of complex numbers
$\displaystyle a+bi=c+di\iff a=c,\ b=d$
Multiplying
$\displaystyle (a+bi)(c+di)$ $\displaystyle =$ $\displaystyle (ac-bd)+(ad+bc)i$
The conjugate
$\displaystyle \bar{z}=a-bi$
A number times its conjugate
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle a^{2}+b^{2}$
Dividing – times the denominator’s conjugate
$\displaystyle \frac{z_{1}}{z_{2}}$ $\displaystyle =$ $\displaystyle \frac{z_{1}\cdot\bar{z}_{2}}{z_{2}\cdot\bar{z}_{2}}$

The modulus and the conjugate

The modulus – the distance from the origin
$\displaystyle |z|$ $\displaystyle =$ $\displaystyle \sqrt{a^{2}+b^{2}}$
The distance between two points
$\displaystyle |z_{1}-z_{2}|$ $\displaystyle =$ $\displaystyle \sqrt{(a_{1}-a_{2})^{2}+(b_{1}-b_{2})^{2}}$
The product with the conjugate and the modulus
$\displaystyle z\cdot\bar{z}$ $\displaystyle =$ $\displaystyle |z|^{2}$
The modulus of a product
$\displaystyle |z_{1}\cdot z_{2}|$ $\displaystyle =$ $\displaystyle |z_{1}|\cdot|z_{2}|$
The conjugate of a sum
$\displaystyle \overline{z_{1}+z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}+\bar{z}_{2}$
The conjugate of a product
$\displaystyle \overline{z_{1}\cdot z_{2}}$ $\displaystyle =$ $\displaystyle \bar{z}_{1}\cdot\bar{z}_{2}$
A number plus its conjugate
$\displaystyle z+\bar{z}$ $\displaystyle =$ $\displaystyle 2\operatorname{Re}(z)$
The triangle inequality
$\displaystyle |z_{1}+z_{2}|\le|z_{1}|+|z_{2}|$

The quadratic equation

The quadratic formula
$\displaystyle z_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$
The sum of the roots
$\displaystyle z_{1}+z_{2}$ $\displaystyle =$ $\displaystyle -\frac{b}{a}$
The product of the roots
$\displaystyle z_{1}\cdot z_{2}$ $\displaystyle =$ $\displaystyle \frac{c}{a}$

The polar form (angles in radians)

The polar form
$\displaystyle z=r(\cos\theta+i\sin\theta)$ $\displaystyle r=|z|$
The notation cis
$\displaystyle r\operatorname{cis}\theta$ $\displaystyle =$ $\displaystyle r(\cos\theta+i\sin\theta)$
The real part
$\displaystyle a=r\cos\theta$
The imaginary part
$\displaystyle b=r\sin\theta$
The angle – by the quadrant of the point
$\displaystyle \tan\theta=\frac{b}{a}$ $\displaystyle a\neq0$
A product in polar form
$\displaystyle z_{1}\cdot z_{2}=r_{1}\cdot r_{2}\left[\cos(\theta_{1}+\theta_{2})+i\sin(\theta_{1}+\theta_{2})\right]$
A quotient in polar form
$\displaystyle \frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\operatorname{cis}(\theta_{1}-\theta_{2})$
De Moivre’s theorem
$\displaystyle \left[R(\cos\varphi+i\sin\varphi)\right]^{n}$ $\displaystyle =$ $\displaystyle R^{n}(\cos n\varphi+i\sin n\varphi)$
The solutions of $\displaystyle {z^n}={R(\cos\varphi+i\sin\varphi)}$
$\displaystyle z_{k}=\sqrt[n]{R}\left[\cos\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)+i\sin\left(\frac{\varphi}{n}+\frac{2k\pi}{n}\right)\right]$ $\displaystyle z^{n}=R(\cos\varphi+i\sin\varphi),\ k=0,1,\ldots,n-1$

Loci and rotation

The circle with centre $\displaystyle z_0$ and radius $\displaystyle R$
$\displaystyle |z-z_{0}|=R$
The perpendicular bisector of the segment $\displaystyle z_1z_2$
$\displaystyle |z-z_{1}|=|z-z_{2}|$
Apollonius’ circle
$\displaystyle |z-z_{1}|=m|z-z_{2}|$ $\displaystyle m>0,\ m\neq1$
Rotating by $\displaystyle \theta$ about $\displaystyle z_0$
$\displaystyle w=z_{0}+(z-z_{0})\operatorname{cis}\theta$