What is the name for $|z|$??
The modulus of $z$.
What is the value of $|z|$, where $z = a + bi$??
$$ \sqrt{a^2 + b^2} $$
What is the value of $|z|^2$, where $z = a + bi$??
$$ a^2 + b^2 $$
If $|z|^2 = a^2 + b^2$, how could you also write $|z|^2$??
$$ (a + bi)(a - bi) $$
What is the word definition of the modulus of $z$??
The distance to $z$ from the origin.
What is the name for $\text{arg} z$??
The argument of $z$.
What is the word definition of the argument of $z$??
The angle that a line drawn to $z$ makes with the real axis, in the anticlockwise
What is the range of the argument $\theta$ of a complex number??
$$ -\pi < \theta \le \pi $$
What typically are the units of $\text{arg} z$??
Radians.
What is the principal argument??
The argument of $z$ in the range $-\pi < \theta \le \pi$.
Assuming all points are some horizontal distance $a$ away from the origin and a vertical distance $b$ in 
, how could you write the angle they make with the real axis??
$$ \tan^{-1}\left(\frac{b}{a}\right) $$
In , what is the formula in terms of $\alpha$ for the argument of $z$ for the first quadrant??
$$ \text{arg}z = \alpha $$
In , what is the formula in terms of $\alpha$ for the argument of $z$ for the second quadrant??
$$ \text{arg}z = \pi-\alpha $$
In , what is the formula in terms of $\alpha$ for the argument of $z$ for the third quadrant??
$$ \text{arg}z = -(\pi-\alpha) $$
In , what is the formula in terms of $\alpha$ for the argument of $z$ for the fourth quadrant??
$$ \text{arg}z = -\alpha $$
Visualise the 4 quadrants of an Argand diagram??
What is the argument of $3 + 4i$, in terms of $\tan$??
$$ \tan^{-1}\left(\frac{4}{3}\right) $$
What quadrant does $3 + 4i$ lie in??
The first quadrant.
What is the argument of $-3 + 4i$, in terms of $\tan$??
$$ \pi - \tan^{-1}\left(\frac{4}{3}\right) $$
What quadrant does $3 - 4i$ lie in??
The second quadrant.
What is the argument of $-3 - 4i$, in terms of $\tan$??
$$ -(\pi - \tan^{-1}\left(\frac{4}{3}\right)) $$
What quadrant does $-3 - 4i$ lie in??
The third quadrant.
What is the argument of $3 - 4i$, in terms of $\tan$??
$$ -\tan^{-1}\left(\frac{4}{3}\right) $$
What quadrant does $3 - 4i$ lie in??
The fourth quadrant.
What quadrant does $12 + 5i$ lie in??
The first quadrant.
What quadrant does $-3 + 6i$ lie in??
The second quadrant.
What quadrant does $-8 -7i$ lie in??
The third quadrant.
What quadrant does $2 - 2i$ lie in??
The third quadrant.
For $a + bi$, why shouldn’t you put the signs of $a$ and $b$ in $\tan\left(\frac{b}{a}\right)$??
Because you’re calculating the angle from a triangle, the lengths of the sides can’t be negative.
If $z = a + bi$, how could you write $a$ in terms of $r$ and $\theta$??
$$ a = r\cos\theta $$
If $z = a + bi$, how could you write $b$ in terms of $r$ and $\theta$??
$$ b = r\sin\theta $$
The rule that $a = r\cos\theta$ and $b = r\sin\theta$ is similar to what in Physics??
$$ F_x = F\cos\theta $$ $$ F_y = F\sin\theta $$
What do you get if you substitute $a = r\cos\theta$ and $b = r\sin\theta$ into $a + bi$??
$$ z = r\cos\theta + ri\sin\theta $$ $$ z = r(\cos\theta + i\sin\theta) $$
What is the $\sin$ and $\cos$ form of $z$??
$$ z = r(\cos\theta + i\sin\theta) $$
What is $r\cos\theta$??
$$ a $$
What is $r\sin\theta$??
$$ b $$
How can you rewrite $|z_1 z_2|$??
$$ |z_1||z_2| $$
How can you rewrite $\text{arg}(z_1 z_2)$??
$$ \text{arg}(z_1) + \text{arg}(z_2) $$
How can you rewrite $z_1 z_2$ in polar form??
$$ r_1 r_2 ( \cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)) $$
How can you rewrite $|\frac{z_1}{z_2}|$??
$$ \frac{|z_1|}{|z_2|} $$
How can you rewrite $\text{arg}(\frac{z_1}{z_2})$??
$$ \text{arg}(z_1) - \text{arg}(z_2) $$
How can you rewrite $\frac{z_1}{z_2}$ in polar form??
$$ \frac{r_1}{r_2} ( \cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)) $$
How can you rewrite $4(\cos(90^{\circ}) - i\sin(90^{\circ}))$??
$$ 4(\cos(-90^{\circ}) + i\sin(-90^{\circ})) $$
Why is $4(\cos(90^{\circ}) - i\sin(90^{\circ}))$ not valid polar form??
Because there is a minus in front of the $\sin$.
How can you rewrite $(\cos(\theta) - i\sin(\theta))$??
$$ (\cos(-\theta^{\circ}) + i\sin(-\theta^{\circ})) $$
Fixing $(\cos(\theta) - i\sin(\theta))$ relies on what property of $\sin$??
$$ \sin(\theta) = -\sin(-\theta) $$
Why is $\frac{16}{3} \pi$ not a valid argument in polar form??
Because it’s not in the range $-pi < \theta \le \pi$.
How could you fix something like $\frac{16}{3} \pi$??
Keep on subtracting $2\pi$ until it’s in the range $-pi < \theta \le \pi$.
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Metadata
date: 2020-11-05 17:00
tags:
- '@?further-maths'
- '@?public'
- '@?complex-numbers'
title: Further Maths - Polar Form