Questions: Find the rectangular form of the complex number (z). Use whatever identities are necessary to find the exact values. The exercise should be worked out without the aid of a calculator. (z=3 operatornamecisleft(frac3 pi4right)=a+b i text where ) (a=square) (b=square)

$Find the rectangular form of the complex number (z). Use whatever identities are necessary to find the exact values. The exercise should be worked out without the aid of a calculator. (z=3 operatornamecisleft(frac3 pi4right)=a+b i text where ) (a=square) (b=square)$

Transcript text: Find the rectangular form of the complex number $z$. Use whatever identities are necessary to find the exact values. The exercise should be worked out without the aid of a calculator. \[ \begin{array}{l} z=3 \operatorname{cis}\left(\frac{3 \pi}{4}\right)=a+b i \text { where } \\ a=\square \\ b=\square \end{array} \]

Solution

Solution Steps

To convert the complex number from polar form to rectangular form, we use the identity $ z = r \operatorname{cis}(\theta) = r(\cos(\theta) + i\sin(\theta)) $. Here, $ r = 3 $ and $ \theta = \frac{3\pi}{4} $. We need to calculate $ a = r \cos(\theta) $ and $ b = r \sin(\theta) $.

Step 1: Convert Polar to Rectangular Form

We start with the complex number in polar form given by $ z = 3 \operatorname{cis}\left(\frac{3\pi}{4}\right) $. Using the identity $ z = r(\cos(\theta) + i\sin(\theta)) $, we identify $ r = 3 $ and $ \theta = \frac{3\pi}{4} $.

Step 2: Calculate $ a $ and $ b $

Next, we calculate the rectangular components $ a $ and $ b $: \[ a = r \cos\left(\frac{3\pi}{4}\right) = 3 \cos\left(\frac{3\pi}{4}\right) = 3 \left(-\frac{\sqrt{2}}{2}\right) = -\frac{3\sqrt{2}}{2} \] \[ b = r \sin\left(\frac{3\pi}{4}\right) = 3 \sin\left(\frac{3\pi}{4}\right) = 3 \left(\frac{\sqrt{2}}{2}\right) = \frac{3\sqrt{2}}{2} \]

Step 3: Final Rectangular Form

Thus, the rectangular form of the complex number $ z $ is: \[ z = a + bi = -\frac{3\sqrt{2}}{2} + \frac{3\sqrt{2}}{2} i \]