4.1: Polar Coordinates
Everyone has dreamed of flying at one time or another. Not only would there be much less traffic to worry about, but directions would be so much simpler!
Walking or driving: "Go East 2 blocks, turn left, then North 6 blocks. Wait for the train. Turn right, East 3 more blocks, careful of the cow! Turn left, go North 4 more blocks and park."
Flying: "Fly 30deg East of North for a little less than 11 and 1/4 blocks. Land."
Nice daydream, what does it have to do with polar coordinates?
Watch This
Embedded Video:
 Khan Academy: Polar Coordinates 1
Guidance
The Polar Coordinate System is alternative to the Cartesian Coordinate system you have used in the past to graph functions. The polar coordinate system is specialized for visualizing and manipulating angles.
Angles are identified by travelling counterclockwise around the circular graph from the 0deg line, or raxis (where the + x axis would be) to a specified angle.
To plot a specific point, first go along the raxis by r units. Then, rotate counterclockwise by the given angle, commonly represented "θ". Be careful to use the correct units for the angle measure (either radians or degrees).
Radians
Usually polar plots are done with radians (especially if they include trigonometric functions), but sometimes degrees are used.
A radian is the angle formed between the r axis and a polar axis drawn to meet a section of the circumference that is the same length as the radius of a circle.
Given that the circumference of a circle is
If 1/2 of a circle is
That translates to approximately 57.3 degrees = 1 radian.
Graphing using technology
Polar Equations can be graphed using a graphing calculator: With the graphing calculator go to MODE. There select RADIAN for the angle measure and POL (for Polar) on the FUNC (function)line. When Y = is pressed, note that the equation has changed from y = to r = . There input the polar equation. After pressing graph, if you can’t see the full graph, adjust x and y max/min, etc in WINDOW.
Example A
Plot the points on a polar coordinate graph:
Point A
Point B
Point C
Solution
Below is the pole, polar axis and the points A, B and C.
Example B
Plot the following points:
a.
b.
c.
d.
e.
Solution
Example C
Use a graphing calculator or plotting program to plot the following equations:
a.
b.
Solution
a.
b.
Review the steps above under Graphing using technology if you are having trouble.
Vocabulary
The polar coordinate system is a specialized graph used for angles and angle manipulations.
The pole is the center point on a polar graph.
One radian is the angle formed by moving counterclockwise around the circumference of a circle by the length of the radius. It is equal to apx 57.3 degrees.
The polar axis is a ray drawn from the pole at the 0^{o} angle on a polar graph.
Guided Practice
1) Plot the points on a polar graph:

a)
(2,π3) 
b)
(3,90o) 
c)
(1.5,π)
2) Convert from radians to degrees:

a)
π2 
b)
5.17 
c)
3π2
3) Convert from degrees to radians:

a)
251o 
b)
360o 
c)
327o
4) Convert from degrees to radians, answer in terms of

a)
90o 
b)
270o 
c)
45o
Answers
1) The points are plotted on the graph below:
2) Recall that

a) If
πrad=180o thenπ2rad=90o 
b) If
1rad≈57.3o then5.17rad≈296o 
c) If
πrad=180o then3π2rad=270o
3) Recall that

a) If
57.3o≈1rad then251o≈4.38rad≈1.4πrad 
b) If
57.3o≈1rad then360o≈6.28rad 
c) If
57.3o≈1rad then327o57.3o≈5.71rad
4) Recall that

a) If
πrad=180o thenπ2rad=90o 
b) If
πrad=180o andπ2rad=90o then112πrad→32π→3π2rad=270o  c) If \begin{align*}\frac{\pi}{2}rad = 90^o\end{align*} then \begin{align*}\frac{\pi}{4} rad = 45^o\end{align*}
Practice
 Why can a point on the plane not be labeled using a unique ordered pair \begin{align*}(r, \theta)\end{align*}
 Explain how to graph \begin{align*}(r, \theta)\end{align*} if \begin{align*}r < 0\end{align*} and/or \begin{align*}\theta > 360\end{align*}
Graph Each Point in the Polar Plane
 A \begin{align*}(6, 145^o)\end{align*}
 B \begin{align*}\left(2, \frac{13\pi}{6} \right)\end{align*}
 C \begin{align*}\left(\frac{7}{4}, 210^o\right)\end{align*}
 D \begin{align*}\left(5, \frac{\pi}{2}\right)\end{align*}
 E \begin{align*}\left(3.5, \frac{\pi}{8}\right)\end{align*}
Name Two Other Pairs of Polar Coordinates for Each Point
 \begin{align*}(1.5, 170^o)\end{align*}
 \begin{align*}\left(5, \frac{\pi}{3}\right)\end{align*}
 \begin{align*}(3, 305^o)\end{align*}
Graph Each Polar Equation
 \begin{align*}r = 3\end{align*}
 \begin{align*}\theta = \frac{\pi}{5}\end{align*}
 \begin{align*}r = 15.5\end{align*}
 \begin{align*}r = 1.5\end{align*}
 \begin{align*}\theta = 175^o\end{align*}
Find the Distance Between Points
 \begin{align*}P_1 \left(5, \frac{\pi}{2}\right)\end{align*} and \begin{align*}P_2 \left(7, \frac{3\pi}{9}\right)\end{align*}
 \begin{align*}P_1 (1.3, 52^o) \end{align*} and \begin{align*}P_2 (13.6, 162^o)\end{align*}
 \begin{align*}P_1 (3, 250^o) P_2 (7, 90^o)\end{align*}
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Cartesian coordinate system
The Cartesian plane is a grid formed by a horizontal number line and a vertical number line that cross at the (0, 0) point, called the origin.polar axis
The polar axis is a ray drawn from the pole at the angle on a polar graph.polar coordinate system
The polar coordinate system is a special coordinate system in which the location of each point is determined by its distance from the pole and its angle with respect to the polar axis.pole
The pole is the center point on a polar graph.radian
A radian is a unit of angle that is equal to the angle created at the center of a circle whose arc is equal in length to the radius.Image Attributions
Here you will learn about the polar coordinate system, which is similar in some ways to the (x, y) graphs you have worked with in the past, but is specialized for visually exploring angles.