10.3 Polar Coordinates

Write the polar coordinates $\left(- 1 , \frac{2 \pi}{3}\right)$ as rectangular coordinates.

Rewrite the Cartesian equation $y = 3 x + 2$ as a polar equation.

Rewrite the Cartesian equation $y^{2} = 3 - x^{2}$ in polar form.

Rewrite the polar equation $r = 2 \sin \theta$ in Cartesian form.

Rewrite the polar equation $r = \sin \left(2 \theta\right)$ in Cartesian form.

How are polar coordinates different from rectangular coordinates?

Explain how polar coordinates are graphed.

Explain why the points $\left(- 3 , \frac{\pi}{2}\right)$ and $\left(3 , - \frac{\pi}{2}\right)$ are the same.

Convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining $\theta$ for the point.

$\left(5 , \pi\right)$

Convert the given polar coordinates to Cartesian coordinates. Remember to consider the quadrant in which the given point is located when determining $\theta$ for the point.

$\left(- 3 , \frac{\pi}{6}\right)$

Convert the given Cartesian coordinates to polar coordinates with $r > 0 , 0 \leq \theta < 2 \pi .$ Remember to consider the quadrant in which the given point is located.

$\left(4 , 2\right)$

Convert the given Cartesian coordinates to polar coordinates with $r > 0 , 0 \leq \theta < 2 \pi .$ Remember to consider the quadrant in which the given point is located.

$\left(3 , −5\right)$

Convert the given Cartesian coordinates to polar coordinates with $r > 0 , 0 \leq \theta < 2 \pi .$ Remember to consider the quadrant in which the given point is located.

$\left(8 , 8\right)$

Convert the given Cartesian equation to a polar equation.

$y = 4$

Convert the given Cartesian equation to a polar equation.

$y = 2 x^{4}$

Convert the given Cartesian equation to a polar equation.

$x^{2} + y^{2} = 3 x$

Convert the given Cartesian equation to a polar equation.

$x^{2} - y^{2} = 3 y$

Convert the given Cartesian equation to a polar equation.

$x^{2} = 9 y$

Convert the given Cartesian equation to a polar equation.

$9 x y = 1$

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r = 4 \cos \theta$

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r = \frac{6}{\cos \theta + 3 \sin \theta}$

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r = 3 \csc \theta$

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r^{2} = 4 \sec \theta \csc \theta$

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r^{2} = 4$

Convert the given polar equation to a Cartesian equation. Write in the standard form of a conic if possible, and identify the conic section represented.

$r = \frac{3}{\cos \theta - 5 \sin \theta}$

Use a graphing calculator to find the rectangular coordinates of $\left(2 , - \frac{\pi}{5}\right) .$ Round to the nearest thousandth.

Use a graphing calculator to find the polar coordinates of $\left(- 7 , 8\right)$ in degrees. Round to the nearest thousandth.

Use a graphing calculator to find the polar coordinates of $\left(- 2 , 0\right)$ in radians. Round to the nearest hundredth.

Describe the graph of $r = a \sec \theta ; a < 0.$

Describe the graph of $r = a \csc \theta ; a < 0.$