Question 8 [10 points]

Express the value $z$ below in polar form, and the value $w$ in the form $a+b i$. Use the square root symbol ' $\checkmark$ ' where needed to give an exact value for your answer. Be sure to include parentheses where necessary, e $g$, to distinguish $1 /(2 k)$ from $1 / 2 k$.
\[
\begin{array}{l}
z=\frac{-5}{2}-\frac{5 \sqrt{3}}{2} i=0 \\
w=3 e^{\frac{i 5 \pi}{4}}=0
\end{array}
\]

Step 1 ：Given the complex number \(z = \frac{-5}{2}-\frac{5 \sqrt{3}}{2} i\), we need to express it in polar form. The polar form of a complex number is given by \(r(\cos \theta + i \sin \theta)\), where \(r\) is the magnitude of the complex number and \(\theta\) is the angle it makes with the positive real axis.

Step 2 ：We calculate the magnitude \(r\) as \(\sqrt{(-\frac{5}{2})^2 + (-\frac{5 \sqrt{3}}{2})^2} = 5\).

Step 3 ：We calculate the angle \(\theta\) as \(\arctan\left(\frac{-\frac{5 \sqrt{3}}{2}}{-\frac{5}{2}}\right) = -\frac{2\pi}{3}\).

Step 4 ：So, the polar form of \(z\) is \(5e^{-\frac{2\pi}{3}i}\).

Step 5 ：Given the complex number \(w = 3 e^{\frac{i 5 \pi}{4}}\), we need to express it in the form \(a+bi\). The rectangular form of a complex number is given by \(r\cos \theta + i r\sin \theta\), where \(r\) is the magnitude of the complex number and \(\theta\) is the angle it makes with the positive real axis.

Step 6 ：We calculate the real part \(a\) as \(3\cos\left(\frac{5 \pi}{4}\right) = -\frac{3\sqrt{2}}{2}\).

Step 7 ：We calculate the imaginary part \(b\) as \(3\sin\left(\frac{5 \pi}{4}\right) = -\frac{3\sqrt{2}}{2}\).

Step 8 ：So, the rectangular form of \(w\) is \(-\frac{3\sqrt{2}}{2} - \frac{3\sqrt{2}}{2}i\).

Step 9 ：Final Answer: \(\boxed{z = 5e^{-\frac{2\pi}{3}i}, w = -\frac{3\sqrt{2}}{2} - \frac{3\sqrt{2}}{2}i}\)