1. Find the Radius ($r$): $r = \sqrt{a^2 + b^2}$. (The Magnitude).
2. Find the Angle ($ heta$): $\theta = \tan^{-1}(b/a)$. (The Direction).
3. Check the Quadrant: If the Real part ($a$) is negative, add $180^\circ$ to your angle.
4. Write in cis: $z = r \text{ cis } \theta$.
Calculate the modulus $|z|$ for each complex number.
$z = 3 + 4i$
$z = 5 - 12i$
$z = -8 + 0i$ (Purely Real).
Find the argument $\theta$ in degrees ($0^° \le \theta < 360^°$).
$z = 1 + i$
$z = -1 + i$
$z = 0 + 10i$ (Purely Imaginary).
If $z = -5 - 5i$, what is the Real part? What is the Imaginary part? Which way are you pointing? If you use a calculator, it will tell you $45^\circ$. Why is this answer incomplete? What must you do to reach the 3rd quadrant?
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Convert $z = 4\sqrt{3} + 4i$ to Polar Form.
Given $z_1 = 2 \text{ cis } 30^\circ$ and $z_2 = 5 \text{ cis } 60^\circ$.
Task: Find the product $z_1 \cdot z_2$ in Polar form.
(Hint: Multiply the radii and add the angles).
Question: In what direction is the combined spirit pointing?
Objective: Explain Polar Form to a younger student using a treasure map.
The Activity:
1. Draw a big 'X' on a map.
2. Give them two instructions: "Walk 10 steps toward the Big Tree."
3. The '10 steps' is the Magnitude, and the 'Tree' is the Angle.
The Lesson: "In math, we don't just say 'How far', we say 'Which way'. We want to always point our steps toward the Father."
Response: ___________________________________________________________