Exam 8: Polar Coordinates and Parametric Equations

Convert the polar equation $r ^ { 2 } = \sin 2 \theta$ to rectangular form.

Sketch a graph of the rectangular equation. [Hint: First convert the equation to polar coordinates.] $\left( x ^ { 2 } + y ^ { 2 } + x \right) ^ { 2 } = 16 \left( x ^ { 2 } + y ^ { 2 } \right)$

Convert the equation to polar form. $x ^ { 2 } - y ^ { 2 } = 4$

Write $- 2 \sqrt { 3 } - 2 i$ in polar form, with $\theta$ between $0$ and $2 \pi$ .

For the point with polar coordinates $P ( - 6,0 )$ find two other polar coordinate representations of P with $r < 0 .$

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

Find $( 3 - \sqrt { 3 } i ) ^ { 3 }$ using DeMoivre's Theorem.

Sketch the graph of the polar equation. $r = 6 \cos 3 \theta$

Use a graphing device to draw the curve represented by $x = 2 \cos 2 t + t$ , $y = 2 \sin 2 t + t$ .

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

Use a graphing device to sketch the curve represented by $x = \sin 5 t , y = \cos 2 t$ .

Write $2 + 2 i$ in polar form, with $\theta$ between $0$ and $2 \pi$ .

Plot the point with polar coordinates $( - 2,8 \pi / 9 )$ and give two other polar coordinate representations, one with $r< 0$ , and the other with $r > 0$ .

Convert the equation $x ^ { 4 } - y ^ { 4 } = 1$ to polar form.

Convert the polar equation to rectangular coordinates. $r = 8 \cos \theta$

Convert the equation $x ^ { 2 } + x y + y ^ { 2 } = 4$ to polar form.

Write $z _ { 1 } = 2 i$ and $z _ { 2 } = - 2 + 2 i$ in polar form, and then find $z _ { 1 } z _ { 2 }$ , $z _ { 1 } / z _ { 2 }$ , and $1 / z _ { 1 }$ .

Find $( 3 + \sqrt { 3 } i ) ^ { - 2 }$ using DeMoivre's Theorem.

Sketch a graph of the polar equation. $r= - 3 \cos 3 \theta$

Test the polar equation for symmetry with respect to the polar axis, the pole, and the line $\theta = \pi / 2$ . $r = \frac { 4 } { 3 - 2 \sin \theta }$