# Further Maths - Polar Form

What is the name for $ \vert z \vert $?

The modulus of $z$.

What is the value of $ \vert z \vert $, where $z = a + bi$?

What is the value of $ \vert z \vert ^2$, where $z = a + bi$?

If $ \vert z \vert ^2 = a^2 + b^2$, how could you also write $ \vert z \vert ^2$?

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?

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?

In , what is the formula in terms of $\alpha$ for the argument of $z$ for the first quadrant?

In , what is the formula in terms of $\alpha$ for the argument of $z$ for the second quadrant?

In , what is the formula in terms of $\alpha$ for the argument of $z$ for the third quadrant?

In , what is the formula in terms of $\alpha$ for the argument of $z$ for the fourth quadrant?

Visualise the 4 quadrants of an Argand diagram?

What is the argument of $3 + 4i$, in terms of $\tan$?

What quadrant does $3 + 4i$ lie in?

The first quadrant.

What is the argument of $-3 + 4i$, in terms of $\tan$?

What quadrant does $3 - 4i$ lie in?

The second quadrant.

What is the argument of $-3 - 4i$, in terms of $\tan$?

What quadrant does $-3 - 4i$ lie in?

The third quadrant.

What is the argument of $3 - 4i$, in terms of $\tan$?

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 $\theta$?

If $z = a + bi$, how could you write $b$ in terms of $r$ and $\theta$?

The rule that $a = r\cos\theta$ and $b = r\sin\theta$ is similar to what in Physics?

What do you get if you substitute $a = r\cos\theta$ and $b = r\sin\theta$ into $a + bi$?

What is the $\sin$ and $\cos$ form of $z$?

What is $r\cos\theta$?

What is $r\sin\theta$?

How can you rewrite $ \vert z _ 1 z _ 2 \vert $?

How can you rewrite $\text{arg}(z _ 1 z _ 2)$?

How can you rewrite $z _ 1 z _ 2$ in polar form?

How can you rewrite $ \vert \frac{z _ 1}{z _ 2} \vert $?

How can you rewrite $\text{arg}(\frac{z _ 1}{z _ 2})$?

How can you rewrite $\frac{z _ 1}{z _ 2}$ in polar form?

How can you rewrite $4(\cos(90^{\circ}) - i\sin(90^{\circ}))$?

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 $(\cos(\theta) - i\sin(\theta))$?

Fixing $(\cos(\theta) - i\sin(\theta))$ relies on what property of $\sin$?

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 $-pi < \theta \le \pi$.