Chapter 9

Sketch a graph of the polar equation. $$ r=\sqrt{3}-2 \sin \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 2 \sqrt{3}-2 i $$

Find the rectangular coordinates for the point whose polar coordinates are given. $$ (\sqrt{3},-5 \pi / 3) $$

Sketch a graph of the polar equation. $$ r=2+\sin \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -1+i $$

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (-1,1) $$

Sketch a graph of the polar equation. $$ r=\sqrt{3}+\cos \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -3 i $$

Show by eliminating the parameter \(\theta\) that the following parametric equations represent a hyperbola: $$ x=a \tan \theta \quad y=b \sec \theta $$

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (3 \sqrt{3},-3) $$

Sketch a graph of the polar equation. $$ r=1-2 \cos \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -3-3 \sqrt{3} i $$

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (\sqrt{8}, \sqrt{8}) $$

Sketch a graph of the polar equation. $$ r^{2}=\cos 2 \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 5+5 i $$

\(37-40=\) Sketch the curve given by the parametric equations. $$ x=t \cos t, \quad y=t \sin t, \quad t \geq 0 $$

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (-\sqrt{6},-\sqrt{2}) $$

Sketch a graph of the polar equation. $$ r^{2}=4 \sin 2 \theta $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 4 $$

\(37-40=\) Sketch the curve given by the parametric equations. $$ x=\sin t, \quad y=\sin 2 t $$

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (3,4) $$

Sketch a graph of the polar equation. $$ r=\theta, \quad \theta \geq 0 \quad(\text { spiral }) $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 4 \sqrt{3}-4 i $$

\(37-40=\) Sketch the curve given by the parametric equations. $$ X=\frac{3 t}{1+t^{3}}, \quad y=\frac{3 t^{2}}{1+t^{3}} $$

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (1,-2) $$

Sketch a graph of the polar equation. $$ r \theta=1, \quad \theta>0 \quad \text { (reciprocal spiral) } $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 8 i $$

\(37-40=\) Sketch the curve given by the parametric equations. $$ x=\cot t, \quad y=2 \sin ^{2} t, \quad 0

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (-6,0) $$

Sketch a graph of the polar equation. $$ r=2+\sec \theta \quad(\text { conchoid }) $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -20 $$

If a projectile is fired with an initial speed of \(v_{0}\) ft \(/ s\) at an angle \(\alpha\) above the horizontal, then its position after \(t\) seconds is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-16 t^{2} $$ (where \(x\) and \(y\) are measured in feet). Show that the path of the projectile is a parabola by eliminating the parameter \(t\)

Convert the rectangular coordinates to polar coordinates with \(r>0\) and \(0 \leq \theta<2 \pi .\) $$ (0,-\sqrt{3}) $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ \sqrt{3}+i $$

Convert the equation to polar form. $$ x=y $$

Use a graphing device to graph the polar equation. Choose the domain of u to make sure you produce the entire graph. $$ r=\cos (\theta / 2) $$

43- 48 . Use a graphing device to draw the curve represented by the parametric equations. $$ x=\sin t, \quad y=2 \cos 3 t $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 3+4 i $$

Convert the equation to polar form. $$ x^{2}+y^{2}=9 $$

Use a graphing device to graph the polar equation. Choose the domain of u to make sure you produce the entire graph. $$ r=\sin (8 \theta / 5) $$

43- 48 . Use a graphing device to draw the curve represented by the parametric equations. $$ x=2 \sin t, \quad y=\cos 4 t $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ i(2-2 i) $$

Convert the equation to polar form. $$ y=x^{2} $$

Use a graphing device to graph the polar equation. Choose the domain of u to make sure you produce the entire graph. $$ r=1+2 \sin (\theta / 2) \quad \text { (nephroid) } $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 3 i(1+i) $$

43- 48 . Use a graphing device to draw the curve represented by the parametric equations. $$ x=3 \sin 5 t, \quad y=5 \cos 3 t $$

Convert the equation to polar form. $$ y=5 $$

Use a graphing device to graph the polar equation. Choose the domain of u to make sure you produce the entire graph. $$ r=\sqrt{1-0.8 \sin ^{2} \theta} \quad \text { (hippopede) } $$

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ 2(1-i) $$

Convert the equation to polar form. $$ x=4 $$