**INTRODUCTION TO POLAR COORDINATES
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to
ingrid.stewart@csn.edu.**

Learning Objectives - This is what you must know after studying the lecture and doing the practice problems!

1. Memorize the definition of the polar coordinate system.

2. Graph points in the polar coordinate system.

3. Change polar coordinates into rectangular coordinates.

4. Change rectangular coordinates into polar coordinates.

So far, we have been representing graphs of equations as a collection of points

in the(x, y)Rectangular (Cartesian) Coordinate System. For example, in the picture below, we see the pointbeing(x, y).(4, 3)Now, we will study a second system called the

Polar Coordinate System. The initial motivation for the introduction of thePolar Coordinate Systemwas the study of circular and orbital motion. Polar coordinates are now used in navigation and for many phenomena in the physical world.

The Polar Coordinate SystemJust like in the rectangular coordinate system we draw a horizontal and a vertical line that intersect at a right angle.The point where the lines intersect is no longer called the Origin, but the

Pole.Instead of rectangular gridlines, we now draw angles

with their vertices at thePoleand their initial side at thepolar axis. Their terminal sides are called. Equally spaced circles represent the "grid lines."r-axesA point in the

Polar Coordinate Systemis represented by the polar coordinates . The numberlies on anrr-axiswhich is the terminal side of angle. For example, in the picture below, we see the pointbeing.(4, 45^{o})

NOTES1. The

r-axis is positive along the terminal side of the angle and negative along the EXTENSION of the terminal side through the pole.2.

and can be positive or negative.r3. can be expressed in radians or degrees.

Strategy for Graphing Pointsin the Polar Coordinate Systems - see #1 and 2in the "Examples" document

- Starting at the
polar axis, draw the angleto find its terminal side.This is the positiver-axis.- Next, extend the terminal side of the angle
through the pole.This is the negativer-axis.- If
is positive, plot the point along the positiverr-axis.- If
is negative, plot the point along the negativerr-axis.In a

Rectangular Coordinate System, there exists exactly one set of coordinates for each point. On the other hand, in aPolar Coordinate System, a point has infinitely many sets of coordinates.

Formulas for Changing Rectangular Coordinates into Polar Coordinates and Vice VersaIt is often necessary to transform from rectangular to polar form or vice versa. The following relationships are useful in this regard.

NOTE: YOU MUST MEMORIZE THESE RELATIONSHIPS!

and the standard form of the equation of a circle with center at the origin:

They are derived from the following picture:

Strategy for Changing Polar Coordinates into Rectangular Coordinates - see #3 and 4in the "Examples" document

- Sketch the point in the
Polar Coordinate Systemto give you an idea of its location.- Use the conversions and to find the rectangular coordinates.

Strategy for Changing Rectangular Coordinates into Polar Coordinates - see #5 through 8in the "Examples" document

- Sketch the point
in the(x, y)Rectangular Coordinate Systemto give you an idea of its location.- Use the conversion to find the
r-value. Depending on the instructions, use either the positive or negative r-value.- Use the conversion to find the angle . Depending on the instructions and/or the location of the point, you may have to work with the
Reference Angleto find the appropriate . Don't forget thatQuadrantal Anglesdo not haveReference Angles!