The Complex Plane; De Moivre's
Theorem
1. Write the Complex Number in Polar Form. SOLUTION
(a) \( z= -2 + 2i \) (b) \(z=\sqrt 3
- i \)
2. Let \( \displaystyle z = 3 \left( \cos \frac{\pi}{12} + i \sin
\frac{\pi}{12} \right), w=6 \left(\cos\frac{\pi}{6} + i \sin
\frac{\pi}{6} \right)\). Find (a) \(
\displaystyle zw\), and (b) \( \displaystyle \frac{z}{w}\).
SOLUTION
3. Let \( \displaystyle z = 2 \left( \cos
\frac{7\pi}{12} + i \sin \frac{7\pi}{12} \right), w=8
\left(\cos\frac{\pi}{4} + i \sin \frac{\pi}{4} \right)\). Find (a) \(
\displaystyle zw\), and (b) \( \displaystyle \frac{z}{w}\).
SOLUTION
4. Write the expression in the standard form \( a+bi\).
SOLUTION
\(\displaystyle \left[ \sqrt 3 (\cos 10^\circ + i \sin 10^\circ)
\right]^6 \)
5. Write the expression in the standard form \(a+bi\).
SOLUTION
\( \displaystyle \left[ \sqrt 5 \left(\cos
\frac{5\pi}{16}+ i \sin \frac{5\pi}{16}\right) \right]^4\)