# Further Maths - Polar Form

> Source: https://ollybritton.com/notes/a-level/further-maths/topics/polar-form/ · Updated: 2020-11-05 · Tags: further-maths, complex-numbers

##### 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 ![PHOTO](argand-diagram-quadrants.png), how could you write the angle they make with the real axis??
$$
\tan^{-1}\left(\frac{b}{a}\right)
$$

##### In ![PHOTO](argand-diagram-quadrants.png), what is the formula in terms of $\alpha$ for the argument of $z$ for the first quadrant??
$$
\text{arg}z = \alpha
$$

##### In ![PHOTO](argand-diagram-quadrants.png), what is the formula in terms of $\alpha$ for the argument of $z$ for the second quadrant??
$$
\text{arg}z = \pi-\alpha
$$

##### In ![PHOTO](argand-diagram-quadrants.png), what is the formula in terms of $\alpha$ for the argument of $z$ for the third quadrant??
$$
\text{arg}z = -(\pi-\alpha)
$$

##### In ![PHOTO](argand-diagram-quadrants.png), 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??
![PHOTO](argand-diagram-quadrants.png)

##### 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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