1. Power the Radius: New Radius = $r^n$. (Exponential Growth).
2. Multiply the Angle: New Angle = $n \cdot \theta$. (Circular Rhythm).
3. Simplify the Loop: If the angle is $> 360$, subtract multiples of 360.
4. Rectify (Optional): Convert back to $a + bi$ if asked.
Calculate the power of each complex number using De Moivre's Theorem.
$(2 \text{ cis } 30^\circ)^3$
$(3 \text{ cis } 40^\circ)^2$
$(\sqrt{2} \text{ cis } 45^\circ)^4$
Find the square root of each complex number ($n = 1/2$).
$z = 16 \text{ cis } 60^\circ$
$z = 100 \text{ cis } 180^\circ$
In Part I, Problem 1, you got $8i$ as the result. Convert $2 \text{ cis } 30^\circ$ back to rectangular form ($a+bi$) and FOIL it three times. Does it match $8i$? Why is De Moivre's Theorem so much more efficient for the "Mystic" walker?
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The Seven-Fold Loop: Calculate $(1 \text{ cis } 60^\circ)^7$.
1. Multiply angle: $60 \times 7 = 420$.
2. Subtract a loop: $420 - 360 = ...$
Final Position: **1 cis ...**
The number $1$ can be written as $1 \text{ cis } 0^\circ$.
It can also be written as $1 \text{ cis } 360^\circ$.
Task: Find the three **cube roots** of 1 ($\sqrt[3]{1}$).
Hint: Divide $0^\circ, 360^\circ,$ and $720^\circ$ by 3.
Plot these three points. What shape do they form?
Objective: Explain De Moivre's Theorem to a younger student using a swing.
The Activity:
1. Push them once. they go to a certain height.
2. Push them again at the right time. they go higher.
3. "Each push is a Power. the more we push, the bigger the radius of the swing gets!"
The Lesson: "God multiplies our little pushes of faith to take us higher and higher until we can see over the trees."
Response: __________________________________________________________