Polar Coordinates And Equations
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.. image:: reflections/9.png
   :alt: Reflection 9 would go here


How to graph polar coordinates
------------------------------

**Source**: 

**Explanation**: 

This artifact demonstrates how to graph polar coordinates.

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

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

**Artifact**:

Plot the point :math:`P(2, { \pi \over 3})`

.. image:: graphs/9_graph_coord.png

Converting polar coordinates to rectangular coordinates and rectangular to polar
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**Source**: Notes, Section 6.4 example 3A

**Explanation**: 

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

#. In this example I use the equation :math:`r^2 = x^2 + y^2` to solve for the directed distance (x) and :math:`tan^{-1}({y \over x})` to solve for the directed angle (y).

#. In this example I use the formulas :math:`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**:

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

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

Polar coordinate = :math:`(\sqrt{53}, 74^\circ)`

#. Convert the polar coordinate :math:`(3, {5\pi \over 6})` into a rectangular coordinate.

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

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

Rectangular coordinate = :math:`(-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.

:math:`r^2 = x^2 + y^2\\
x = r \cos \theta\\
y = r \sin \theta`

#. Here I simplify :math:`r = 4 \sec \theta` into :math:`r \cos \theta` so I can substitute for :math:`x` using the formula :math:`x = r \cos \theta`. The answer is the line :math:`x = 4`.

#. 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 :math:`\\x = r \cos \theta \text{ and } y = r \sin \theta` to convert them from rectangular form into polar form.

**Artifact**:

#. Convert :math:`r = 4 \sec \theta` to rectangular form.

.. math::

   r = 4 \sec \theta

   {r \over \sec \theta} = 4

   r \cos \theta = 4

   x = 4

#. Convert :math:`(x-3)^2 + (y-2)^2 = 13` to polar form.

.. math::

   (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 :math:`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.

.. image:: graphs/9_graph.png
   :height: 500px
   :width: 700 px

The rest of the information can be gathered visually.

**Artifact**:

Analyze the graph of :math:`r = 3 -3 \sin \theta`.

Domain: All real numbers

Range: [0, 6]

Symmetric about the y-axis

Maximum r-value = 6