91Ó°ÊÓ

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=3 e^{i \frac{13 \pi}{18}}\) \(w=4 e^{i \frac{3 \pi}{2}}\)

Identify and graph each polar equation. $$ r=2+2 \cos \theta $$

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ (2, \pi) $$

Identify and graph each polar equation. $$ r=1+\sin \theta $$

Find \(z w\) and \(\frac{z}{w} .\) Write each answer in polar form and in exponential form. \(z=2 e^{i \frac{4 \pi}{9}}\) \(w=6 e^{i \frac{10 \pi}{9}}\)

Suppose that \(\mathbf{v}\) and \(\mathbf{w}\) are unit vectors. If the angle between \(\mathbf{v}\) and \(\mathbf{i}\) is \(\alpha\) and the angle between \(\mathbf{w}\) and \(\mathbf{i}\) is \(\beta\), use the idea of the dot product \(\mathbf{v} \cdot \mathbf{w}\) to prove that $$ \cos (\alpha-\beta)=\cos \alpha \cos \beta+\sin \alpha \sin \beta $$

Identify and graph each polar equation. $$ r=3-3 \sin \theta $$

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\). $$ \left(-3,-\frac{\pi}{4}\right) $$

Plot each point given in polar coordinates, and find other polar coordinates \((r, \theta)\) of the point for which: (a) \(r>0, \quad-2 \pi \leq \theta<0\) (b) \(r<0, \quad 0 \leq \theta<2 \pi\) (c) \(r>0, \quad 2 \pi \leq \theta<4 \pi\) $$ \left(-2,-\frac{2 \pi}{3}\right) $$

Identify and graph each polar equation. $$ r=2+\sin \theta $$