Question

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

$4-3i$

Short answer

The polar form of$z=4-3i$is$5(\cos 323.1^o+i\sin 323.1^o)$

Step-by-step solution

Step 1. Given data 

$z=4-3i$

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)=(4, -3),

$r=\sqrt{{\left(4\right)}^2+{(-3)}^2}$

$r=\sqrt{16+9}$

$r=\sqrt{25}$

$r=5$

Step 3. Calculating the value of θ

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

$\sin \theta =\frac{y}{r}=\frac{-3}{5}$and $\cos \theta =\frac{4}{5}$

Since, $\sin 323.1^o=\frac{-3}{5}$and $\cos 323.1^0=\frac{4}{5}$.

Therefore, the polar form of $z=4-3i$is$5(\cos 323.1^o+i\sin 323.1^o)$