8.1+Content+-+Polar+Graphs

toc Add Polar Graph content to this page. See specific details on the home page. This unit has been split into parts A & B. Content between the two must be cohesive as it is one continuous, connected material. They are split simply to better organize the page.
 * Quick Instructions**


 * Polar Graphs (Unit 8A) Learning Targets**
 * Graph functions of polar coordinates using a corresponding rectangular graph.
 * Graph functions with restictions and identify values of theta that form portions of graphs.
 * Quickly sketch the graphs of Circles, Roses, Limaçons and Cardioids.
 * Write equations that describe the relationship between the polar coordinates of graphs of Circles, Roses, Limaçons and Cardioids.
 * Solve systems of equations in polar form for simultaneous and non-simultaneous solutions.

=What is A Polar Graph?=
 * The diagram to the right is a Polar Plane
 * Polar graphs and coordinates are simply another way of plotting a locust of points
 * Instead of the traditional x and y axis of a rectangular coordinate plane, a Polar Plane plots points based on a points radius and angle
 * Polar coordinates are written as (r,θ)
 * "r" stands for radius- it is the distance from the pole to the point
 * θ is the angle formed by a ray starting from the origin and running through the point and the x-axis
 * Each concentric ring represents a new "r" value, increasing as you move away from the center
 * Much like how a regular (x,y) coordinate tells you how far left, right, up, or down, you go A polar coordinate (r,θ) tells you how far from the center you go at a certain angle

= = = = = = = =

=**Polar form to Rectangular form & Rectangular to Polar form**=

Sometimes, the equation or problem you are trying to solve will require you to change the coordinate system that you are working in. One reason would be to properly graph something, or when polar equations won't work for a word problem.

To go from **Polar Coordinates to Rectangular Coordinates**, do the following: math \\ \text{1)\indent Take your polar point, in the form of} \; (r,\theta)}\\ \text{2)\indent To find the x coordinate, evaluate} \;r\cos\theta}\\ \text{3)\indent To find the y coordinate, evaluate} \; r\sin\theta}\\ \text{4)\indent Your final point in rectangular form will be} \;(r\cos\theta, r\sin\theta)}\\ math

The reason this works is that if we graph a point in polar, then create a right triangle going to the point (r,θ), the corresponding distance from the **y** axis to **x**, and the **x** axis to **y** are equal to **rcosθ** and **rsinθ** respectively, as shown in the diagram below:

This diagram applies to both **rectangular to polar**, and **polar to rectangular**





To go from **Rectangular Coordinates to Polar Coordinates**, do the following: math \\ \text{1)} \indent \text{Start with point} \left(x,y \right)\\ \text{2)} \indent r=\sqrt{x^{2}+y^{2}} \\ \text{3)} \indent \Theta =\tan^{-1}\left ( \frac{y}{x} \right )\\ \text{4)} \indent \text{Your point in Polar will be } \left(r,\Theta \right) math

R^2 = y^2 + x^2 y = r sin t x = r cos t using these relationships, you can substitute into polar equations or rectangular equations, and solve from there.
 * To convert EQUATIONS, remember the following: **

=**Graphs of Polar Equations**=
 * polar graphs represent the relationship between an output (radius) and input (angle).
 * often times, the graph forms specific patterns that we can categorize
 * other times, the graphs do not follow any common patterns and as a result, we must plot points in order to find the shape
 * to write the equation for a polar graph, you may be able to use some of the common patterns below

**Rose Graphs**


This is called the Rose graph. A rose graph is described by either of the following polar equations: math r=a\sin \left ( n\Theta \right )\indent or\indent r=a\cos \left ( n\Theta \right ) math "a" and "n" are numbers that modify the rose.
 * "n" modifies the number of petals in the rose
 * If "n" is even, the number of petals is 2n
 * If "n" is odd, the number of petals is still 2n but **only "n" petals are visible** (half of the petals will overlap)
 * "a" describes the maximum radius length of the petals(think of amplitude)
 * the petals are evenly spaced throughout the plane

Where rose graphs start and why they start there

 * sin rose graphs(with no vertical or horizontal shifts) ALWAYS start at the pole
 * cos rose graphs(again with no vertical or horizontal shifts) ALWAYS start at a maximum.

There are two ways you can approach the "why" part
2. Graphically
 * 1) Unit circle
 * on the unit circle, sin(0) is 0 and cos(0) is 1
 * the pole is at r = 0
 * for cos, the graph will always start at whatever the a value is
 * if you graph y = sin(x), at x=0, y also is 0
 * whenever y or r is at 0, that means it is at the pole no matter the x or theta value

**Limacon Graphs**
There are three different types of limacon graphs: loops, cardioids, and dimpled. The difference between these three are the a's and b's of the limacon equations. The limacon equation is:

math r=b\pm a\sin \Theta \indent or \indent r=b\pm a\cos\Theta math math \\ a\pm b : \text{intercepts along axis of symmetry}\\ \pm b : \text{intercepts along non-symmetrical axis} math

Symmetry
It can be helpful to know if a graph displays symmetry. All Limacons have symmetry, but what determines the axis of symmetry and non-symmetrical axis is whether it is a sine or cosine function. Cosine will have an axis of symmetry on the horizontal while the sine graphs will have an axis of symmetry on the vertical.

Loops


This is a loop limacon. In order for a Limacon to be a loop: math math
 * b|<|a|

Once again: math \\ a\pm b : \text{intercepts along axis of symmetry}\\ \pm b : \text{intercepts along non-symmetrical axis} math

Cardioid


The equation for this Cardioid is: math r=2+2\cos\Theta \\ \\ \text{In Cardioids,} |a|=|b| math Because a=b, there will always be an intercept at the origin.

Dimpled Limacon


The general equation for a Dimpled Limacon is: math r=b\pm a\cos\theta or r=b\pm a\sin\theta \\ \text{For Dimpled Limacons,} |b|>|a| math The equation for this Dimpled Limacon is: math r=4+3\cos\theta \\ \\ math

Circles


These are the different ways to write a circle graph in polar form. As shown above, the three ways are: math r=a \indent or \indent r=a\sin\theta \indent or \indent r=a\cos\theta math "a" represents the radius of the circle for the r=a while for the sin and cos graphs, the "a" represents the diameter and the circle's end points are always the origin unless it is shifted.

There is also a way to represent a line in polar form. We know that math y= rcos\theta \\ \text and \\ x= rsin\theta \\ math so if we have the line in rectangular form, we substitute the x and the y, isolate the r, and you have your equation in polar form!

Non-Standard Polar Graphs


If polar graphs are not standard, meaning that they are not typical graphs such as limacons or roses, then is there a way to graph it? The answer is yes. There are two ways that you can do this. The long way is to plot points, but who would want to do that? Instead, we can sketch the rectangular graph to see the values of r that correspond to increasing values of theta. With these values, sketching any polar graph, though inconsistent is possible! The easiest way to graph polar equations that you do not know, is to draw a line graph for them.
 * Establish the amplitude
 * Establish the period
 * Draw a line graph
 * Plot points at 1/4, 1/2, and 3/4 the period, and at the length of one complete cycle.
 * Then draw a polar graph
 * Find the same points as (r,theta) on the graph
 * draw lines

=How To Write a Polar Equation Given Only the Graph=

If you know how to graph polar equations, then you already know how to write a polar equation only knowing the graph. When you read about the rules of the graphs such as limacons, then all you have to do is plug in the values such as a and b. Whether it is sin or cos, it depends on which axis of the equation is the axis of symmetry. The y- axis indicates a sin while the x-axis indicates a cos graph.

=Solving Systems of Polar Equations = In this section we will explore how to solve a system of equations by finding the simultaneous and non-simultaneous solutions.

__**Simultaneous**__ You get simultaneous solutions of a graph by setting the 2 equations equal to each other.

You get non-simultaneous solutions of a graph by graphing both on a polar plane. Then where they cross each other is where the nonsimultaneous solution is.
 * __Non-Simultaneous__**


 * Primary authors of this page (as of 06/02/12):**
 * Tyler Klivickis**
 * Sam Lee**
 * Jerry Pan**
 * Kirk Halverson**
 * Jay Wang**