Advanced examples for multiplying complex numbers in polar form (Pre-Calculus)

📚 Quick Study Guide

• 🧭 Polar Form: A complex number $z = a + bi$ can be represented in polar form as $z = r(\cos \theta + i \sin \theta)$, where $r = |z| = \sqrt{a^2 + b^2}$ is the magnitude (or modulus) and $\theta = \arctan(\frac{b}{a})$ is the argument (or angle).
• 🔢 Multiplication: If $z_1 = r_1(\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2(\cos \theta_2 + i \sin \theta_2)$, then their product is given by: $z_1z_2 = r_1r_2 [\cos(\theta_1 + \theta_2) + i \sin(\theta_1 + \theta_2)]$.
• ➕ Argument Addition: To multiply complex numbers in polar form, multiply their magnitudes and add their arguments.
• 🔄 Converting Back: If needed, convert the resulting polar form back to rectangular form ($a + bi$) by evaluating the cosine and sine functions and distributing the magnitude.
• 📐 Angle Considerations: Ensure the argument $\theta$ is in the correct quadrant based on the signs of $a$ and $b$. Use $\arctan(\frac{b}{a})$ carefully, considering adding or subtracting $\pi$ (or 180 degrees) if 'a' is negative.

🧪 Practice Quiz

1. Question 1:
   What is the product of $z_1 = 2(\cos(30^\circ) + i \sin(30^\circ))$ and $z_2 = 3(\cos(60^\circ) + i \sin(60^\circ))$ in polar form?
   1. $5(\cos(90^\circ) + i \sin(90^\circ))$
   2. $6(\cos(90^\circ) + i \sin(90^\circ))$
   3. $6(\cos(30^\circ) + i \sin(30^\circ))$
   4. $5(\cos(30^\circ) + i \sin(30^\circ))$
2. Question 2:
   Given $z_1 = 4(\cos(\frac{\pi}{4}) + i \sin(\frac{\pi}{4}))$ and $z_2 = \frac{1}{2}(\cos(\frac{\pi}{2}) + i \sin(\frac{\pi}{2}))$, find $z_1z_2$.
   1. $2(\cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}))$
   2. $4(\cos(\frac{3\pi}{4}) + i \sin(\frac{3\pi}{4}))$
   3. $2(\cos(\frac{\pi}{8}) + i \sin(\frac{\pi}{8}))$
   4. $4(\cos(\frac{\pi}{8}) + i \sin(\frac{\pi}{8}))$
3. Question 3:
   Multiply $z_1 = \cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3})$ by $z_2 = 5(\cos(\pi) + i \sin(\pi))$.
   1. $5(\cos(\frac{4\pi}{3}) + i \sin(\frac{4\pi}{3}))$
   2. $5(\cos(\frac{2\pi}{3}) + i \sin(\frac{2\pi}{3}))$
   3. $6(\cos(\frac{4\pi}{3}) + i \sin(\frac{4\pi}{3}))$
   4. $6(\cos(\frac{2\pi}{3}) + i \sin(\frac{2\pi}{3}))$
4. Question 4:
   What is the magnitude of the product of $z_1 = 3(\cos(45^\circ) + i \sin(45^\circ))$ and $z_2 = 2(\cos(15^\circ) + i \sin(15^\circ))$?
   1. 5
   2. 1
   3. 6
   4. 9
5. Question 5:
   If $z_1 = \sqrt{2}(\cos(\frac{\pi}{6}) + i \sin(\frac{\pi}{6}))$ and $z_2 = \sqrt{2}(\cos(\frac{\pi}{3}) + i \sin(\frac{\pi}{3}))$, what is the real part of $z_1z_2$?
   1. 0
   2. -1
   3. 2
   4. 1
6. Question 6:
   Calculate $z_1z_2$ where $z_1 = 7(\cos(20^\circ) + i \sin(20^\circ))$ and $z_2 = (\cos(10^\circ) + i \sin(10^\circ))$. Express your answer in polar form.
   1. $7(\cos(30^\circ) + i \sin(30^\circ))$
   2. $8(\cos(30^\circ) + i \sin(30^\circ))$
   3. $7(\cos(200^\circ) + i \sin(200^\circ))$
   4. $7(\cos(10^\circ) + i \sin(10^\circ))$
7. Question 7:
   Let $z_1 = 5(\cos(\frac{5\pi}{6}) + i \sin(\frac{5\pi}{6}))$ and $z_2 = 2(\cos(\frac{\pi}{6}) + i \sin(\frac{\pi}{6}))$. Find the argument of $z_1z_2$.
   1. $\frac{\pi}{3}$
   2. $\pi$
   3. $\frac{2\pi}{3}$
   4. $\frac{4\pi}{3}$