# Polar Coordinates And Equations¶

## How to graph polar coordinates¶

Source:

Explanation:

This artifact demonstrates how to graph polar coordinates.

In the polar coordinate $$P(2, { \pi \over 3})$$, the directed distance is 2, and the directed angle is $$\pi \over 3$$.

So the pole starts at 2 on the O ray, and swivels out by $$\pi \over 3$$ degrees.

Artifact:

Plot the point $$P(2, { \pi \over 3})$$

## Converting polar coordinates to rectangular coordinates and rectangular to polar¶

Source: Notes, Section 6.4 example 3A

Explanation:

This artifact demonstrates converting polar coordinates to rectangular coordinates and rectangular to polar.

1. In this example I use the equation $$r^2 = x^2 + y^2$$ to solve for the directed distance (x) and $$tan^{-1}({y \over x})$$ to solve for the directed angle (y).
2. In this example I use the formulas $$x = r \cos \theta \text{ and } y = r \sin \theta$$, and my knowledge of the unit circle, to calculate the approximate values of x and y.

Artifact:

1. Convert the rectangular coordinate (2,7) into a polar coordinate.

$$r = \sqrt{2^2 + 7^2} = \sqrt{53}\\ tan^{-1}({7 \over 2}) = 74^\circ$$

Polar coordinate = $$(\sqrt{53}, 74^\circ)$$

1. Convert the polar coordinate $$(3, {5\pi \over 6})$$ into a rectangular coordinate.

$$x = r \cos \theta\\ x = 3 \cos {5 \pi \over 6}\\ x = 3(-{\sqrt{3} \over 2}) \approx -2.60$$

$$y = r \sin\theta\\ y = 3 \sin {5 \pi \over 6}\\ y = 3({1 \over 2}) \approx 1.5$$

Rectangular coordinate = $$(-2.60, 1.5)$$

## Converting polar equations to rectangular equations and rectangular to polar¶

Source: Section 6.4 Example 5 and Example 6

Explanation:

This artifact demonstrates converting polar equations to rectangular equations and rectangular to polar.

$$r^2 = x^2 + y^2\\ x = r \cos \theta\\ y = r \sin \theta$$

1. Here I simplify $$r = 4 \sec \theta$$ into $$r \cos \theta$$ so I can substitute for $$x$$ using the formula $$x = r \cos \theta$$. The answer is the line $$x = 4$$.
2. Here I simplify the original equation into x’s and y’s in both the second and first degree because I can subsitute them with the conversion formulas $$\\x = r \cos \theta \text{ and } y = r \sin \theta$$ to convert them from rectangular form into polar form.

Artifact:

1. Convert $$r = 4 \sec \theta$$ to rectangular form.
$r = 4 \sec \theta$${r \over \sec \theta} = 4$$r \cos \theta = 4$$x = 4$
1. Convert $$(x-3)^2 + (y-2)^2 = 13$$ to polar form.
$(x-3)^2 + (y-2)^2 = 13$$x^2 - 6x + 9 + y^2 -4y + 4 = 13$$x^2 + y^2 -6x -4y = 0$$r^2 - 6r \cos \theta -4r \sin \theta = 0$$r(r - 6 \cos \theta - 4 \sin \theta) = 0$$r = 0 \text{ or } r - 6 \cos \theta - 4 \sin \theta = 0$

## Graphs of polar equations¶

Source: Section 6.5 Example 5

Explanation:

This artifact demonstrates graphs of polar equations.

From graphing the polar equation $$r = 3 -3 \sin x$$ in radian/function mode, I can tell that the maximum r value is 6 because that is the highest y value you can ever get on this particulur sinusoid.

The rest of the information can be gathered visually.

Artifact:

Analyze the graph of $$r = 3 -3 \sin \theta$$.

Domain: All real numbers

Range: [0, 6]