MHB Find the Modulus & Argument of \(wz\)

[Punch]

The complex number w has modulus \(\sqrt{2}\) and argument \(-\frac{3\pi}{4}\), and the complex number \(z\) has modulus \(2\) and argument \(-\frac{\pi}{3}\). Find the modulus and argument of \(wz\), giving each answer exactly.
By first expressing w and \(z\) is the form \(x+iy\), find the exact real and imaginary parts of \(wz\).
I have a problem with finding the argument of \(wz\) and expressing \(w\) and \(z\) in the form \(x+iy\)

 

[Mr Fantastic]

Re: complex numbers

The complex number w has modulus \sqrt{2} and argument -\frac{3\pi}{4}, and the complex number z has modulus 2 and argument -\frac{\pi}{3}. Find the modulus and argument of wz, giving each answer exactly.
By first expressing w and z is the form x+iy, find the exact real and imaginary parts of wz.
I have a problem with finding the argument of wz and expressing w and z in the form x+iy

Review how to multiply two complex numbers when they are written in polar form.

z = r cis(theta). You need to review polar form.

 

[Prove It]

The complex number w has modulus \(\sqrt{2}\) and argument \(-\frac{3\pi}{4}\), and the complex number \(z\) has modulus \(2\) and argument \(-\frac{\pi}{3}\). Find the modulus and argument of \(wz\), giving each answer exactly.
By first expressing w and \(z\) is the form \(x+iy\), find the exact real and imaginary parts of \(wz\).
I have a problem with finding the argument of \(wz\) and expressing \(w\) and \(z\) in the form \(x+iy\)

If you need to solve this problem by converting to Cartesians, then

\[ \displaystyle \begin{align*} w &= \sqrt{2}\left[\cos{\left(-\frac{3\pi}{4}\right)} + i\sin{\left(-\frac{3\pi}{4}\right)}\right] \\ &= \sqrt{2}\left(-\frac{1}{\sqrt{2}} - \frac{i}{\sqrt{2}}\right) \\ &= -1 - i \end{align*} \]

and

\[ \displaystyle \begin{align*} z &= 2\left[\cos{\left(-\frac{\pi}{3}\right)} + i\sin{\left(-\frac{\pi}{3}\right)}\right] \\ &= 2\left(\frac{1}{2} - \frac{i\sqrt{3}}{2} \right) \\ &= 1 - i\sqrt{3} \end{align*} \]

So multiplying them together gives...

\[ \displaystyle \begin{align*} wz &= \left(-1-i\right)\left(1-i\sqrt{3}\right) \\ &= -1 + i\sqrt{3} - i + i^2\sqrt{3} \\ &= \left(-1 - \sqrt{3}\right) + i\left(-1 + \sqrt{3}\right) \end{align*} \]Can you evaluate the modulus and argument of this complex number?