trigonometry - Complex number + trigo : $-1 + tan(3)i$ , find modulus and argument

I have $-1 + \tan(3)i$ and must find its modulus and its argument. I tried to solve it by myself for hours, and then I looked at the answer, but I am still confused with a part of the solution.

Here is the provided solution:
$$\begin{align} z &= -1 + \tan(3)i \\ &= -1 + \frac{\sin(3)}{\cos(3)}i \\ &= \frac1{\left|\cos(3)\right|} ( \cos(3) + i(-1)\sin(3)) \\ &= \frac1{\left|\cos(3)\right|} e^{-3i} \\ &= \frac1{\left|\cos(3)\right|} e^{(2\pi-3)i} \end{align}$$

I don't understand how we get to $$\frac1{\left|\cos(3)\right|}(\cos(3) + i(-1)\sin(3))$$
How did they get this modulus $1/|\cos(3)|$, and the $-1$ in the imaginary part? How did they reorder the previous expression to obtain this?

I also don't see why they developed the last equality. They put $2\pi-3$ instead of $-3$; OK, it is the same, but what was the aim of a such development?

Thanks!

Answer

Let $z = -1 + \tan(3) \ i$. In the complex plane, this would be the point $(-1, \tan(3))$, which has length

$$|z| = \sqrt{(-1)^2 + \tan^2(3)} = \sqrt{1 + \frac{\sin^2(3)}{\cos^2(3)}} = \sqrt{\frac{1}{\cos^2(3)}} \sqrt{\cos^2(3) + \sin^2(3)} = \frac{1}{|\cos(3)|}$$

For the last equality, we used $\sin^2(x) + \cos^2(x) = 1$. Now we want to write

$$z = |z| \ e^{\omega i} = |z| \ (\cos(\omega) + i \sin(\omega))$$

for some $\omega$. It turns out this can be done easily by writing

$$z = \frac{1}{\cos(3)}(-\cos(3) + i \sin(3)) = \frac{1}{|\cos(3)|}(\cos(-3) + i \sin(-3)) = \frac{1}{|\cos(3)|} e^{-3i}$$

Since $-3 \notin [0, 2\pi)$ they decided to add $2 \pi$ to the angle, so that it is inside this interval.