Answer
$-3-4i=5e^{i4.069}$
Work Step by Step
$\text{Solution:}$
Step 1: Write $-3-4i$ in the form $z=x+iy$. $$z=-3-4i$$
Step 2: So in the complex plane, the $x$-coordinate is $-3$ and the $y$-coordinate is $-4$. That is $(x,y)=(-3,-4)$.
Step 3: Find $r$ using the formula: $r=|z|=\sqrt {x^2+y^2}$. $$r=|z|=\sqrt {(-3)^2+(-4)^2}=\sqrt{ 9+16}=\sqrt {25}=5$$
Step 4: Find $\theta$, using the formula: $\theta=\tan^{-1}\frac{y}{x}$. Since point $(-3,-4)$ is in 3rd quadrant (angles from $\pi$ to $\frac{3\pi}{2}$), therefore,
$$\theta=\tan^{-1}\frac{-4}{-3}=\pi+\tan^{-1}\left(\frac{4}{3}\right)=4.06888$$
Step 5: Write in polar form $z=re^{i\theta}$. Here $(r,\theta)=(5,4.069)$.Therefore, the polar form of $-3-4i$ is $$z=re^{i\theta}=5e^{i4.069}$$
