Question

Write each complex number in polar form. Express each argument in degrees.

$-1-i$

Short answer

The polar form of$z=-1-i$is$\sqrt{2}(\cos {225}^o+i\sin {225}^o)$

Step-by-step solution

Step 1. Given data

$z=-1-i$

Step 2. Calculate the distance r

The polar form of $z=x+iy$is $z=r(\cos \theta +i\sin \theta )$

We know,

$\left|z\right|=r=\sqrt{x^2+y^2}$

Substituting the values of (x, y)=(-1, -1),

$r=\sqrt{{(-1)}^2+{(-1)}^2}$

$r=\sqrt{2}$

Step 3. Calculating the value of θ

Substitute $x=r\cos \theta$and $y=r\sin \theta$,

Therefore, $\sin \theta =\frac{y}{r}=\frac{-1}{\sqrt{2}}$and

$\cos \theta =\frac{x}{r}=\frac{-1}{\sqrt{2}}$

since, $\sin {225}^o=\frac{-1}{\sqrt{2}}$and $\cos {225}^o=\frac{-1}{\sqrt{2}}$

Therefore, $\theta ={225}^o$

Therefore, the polar form of$z=-1-i$is$\sqrt{2}(\cos {225}^0+i\sin {225}^0)$