Let’s solve the problem step by step.

The given expression is:

[ \sqrt{2} \left[ \cos\left(\frac{\pi}{6}\right) - i\sin\left(\frac{\pi}{6}\right) \right] \cdot \sqrt{3} \left[ \cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right) \right] ]

Step 1: Express the factors in polar form

The polar form of a complex number is: $r \left[\cos(\theta) + i\sin(\theta)\right]$

First term:

$\sqrt{2} \left[\cos\left(\frac{\pi}{6}\right) - i\sin\left(\frac{\pi}{6}\right)\right]$ This is equivalent to: $\sqrt{2} \text{cis}\left(-\frac{\pi}{6}\right)$ where $\text{cis}(\theta) = \cos(\theta) + i\sin(\theta)$.

Second term:

$\sqrt{3} \left[\cos\left(\frac{3\pi}{4}\right) + i\sin\left(\frac{3\pi}{4}\right)\right]$ This is equivalent to: $\sqrt{3} \text{cis}\left(\frac{3\pi}{4}\right)$

Step 2: Multiply the polar forms

When multiplying two complex numbers in polar form: $r_1 \text{cis}(\theta_1) \cdot r_2 \text{cis}(\theta_2) = (r_1 r_2) \text{cis}(\theta_1 + \theta_2)$

Here:

• $r_1 = \sqrt{2}$, $\theta_1 = -\frac{\pi}{6}$
• $r_2 = \sqrt{3}$, $\theta_2 = \frac{3\pi}{4}$

So: $r_1 r_2 = \sqrt{2} \cdot \sqrt{3} = \sqrt{6}$ $\theta_1 + \theta_2 = -\frac{\pi}{6} + \frac{3\pi}{4} = \frac{-2\pi}{12} + \frac{9\pi}{12} = \frac{7\pi}{12}$

Thus, the product is: $\sqrt{6} \text{cis}\left(\frac{7\pi}{12}\right)$

Step 3: Write the result in expanded polar form

$\sqrt{6} \text{cis}\left(\frac{7\pi}{12}\right) = \sqrt{6} \left[\cos\left(\frac{7\pi}{12}\right) + i\sin\left(\frac{7\pi}{12}\right)\right]$

Final Answer:

The correct option is: $\text{B: } \sqrt{6} \left[\cos\left(\frac{7\pi}{12}\right) + i\sin\left(\frac{7\pi}{12}\right)\right]$

Would you like further clarifications? Here are 5 related questions:

1. How do you multiply complex numbers in rectangular form instead of polar form?
2. What is the geometric interpretation of multiplying two complex numbers in polar form?
3. How do you convert between polar and rectangular forms of complex numbers?
4. Why does the magnitude multiply and the angle add in polar form multiplication?
5. How do you simplify $-\frac{\pi}{6} + \frac{3\pi}{4}$ in terms of radians?

Tip: Always simplify angles to a common denominator when adding or subtracting them in radians!