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# Copy of Copy of Polar Coordinates

A mathematics research project
by

## Robert Holt

on 23 April 2013

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#### Transcript of Copy of Copy of Polar Coordinates

Polar Coordinates Before we can talk about all of the wonderful things that polar coordinates can do for us, we need to understand the basics! Some Definitions... "A point P in the plane has polar coordinates (r, θ) if the line segment OP has length r and the angle that OP makes with the positive axis is θ (measured in a counter clockwise direction)." "This definition requires that r > 0. If r < 0, then we consider the point Q which has polar coordinates (-r, θ). Then the point P has polar coordinates (r, θ) if P is the point on the straight line containing O and Q which is -r units from O on the opposite side of O from Q" http://archives.math.utk.edu/visual.calculus/0/polar.6/ http://brownsharpie.courtneygibbons.org/?p=7 Polar Coordinates
Stolen by Mr. C Put more simply,
a point in polar coordinates
is represented as (r,θ).

θ is the angle that the point is away from the positive x-axis. Let's do some practice problems on finding the polar coordinate of a point. (0,0°) 1 2 3 A B 1) What are the coordinates of point A?

2) What are the coordinates of point B? Hint:
-The red lines here represent the different distances from the origin. Answers:
Point A has coordinates (2,90°)
Point B has coordinates (1,30°) http://en.wikipedia.org/wiki/Polar_coordinate_system http://www.texample.net/tikz/examples/polar-coordinates-template/ (2,90°) (1,30°) So how are polar coordinates and rectangular coordinates related? Aren't rectangular coordinates enough? Why do we need Polar coordinates? http://thetrig.blogspot.com/2011/11/polar-coordinates.html More practice... Now, the axis is not always labeled using degrees, we can also use radians to represent the angle of a point from the positive axis. A B C D What are the polar coordinates of points A, B, C, and D? A (4,3π/4)
B (3,π/3)
C (1,11π/6)
D (3.5,7π/6) (5,135°) (6.5,75°) (5,30°) (1.5,210°) (4.5,180°) (7,210°) (3.5,270°) (5.5,330°) Commonly, rectangular coordinates are used for mapping because they are often the most useful, but not always. Sometimes, the data that you have collected would be more suited to be graphed in polar coordinates to be more easily understood. For example, if you are using radar, or trying to tell a ship where to travel. http://scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf For example, the data collected here is obviously easier to interpret graphically using polar coordinates rather than rectangular coordinates. Note: the angle can be measured in either degrees or radians Now, so far we have only been labeling points using positive values...Is this the only way to do it? NO!! There are different ways to represent points using both positive and negative values. http://en.wikipedia.org/wiki/Polar_coordinate_system Take for example points A and B. Previously we said that their coordinates were A (2,90°) and B (1,30°). These points can also have coordinates A( -2,270°) and B (-1,210°). (1,210°) (-1,210°) This shows you that polar coordinates are not unique like rectangular coordinates. One point can be represented by many different sets of coordinates, just like A is both (2,90°) and (-2,270°) http://scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf http://scidiv.bellevuecollege.edu/dh/ccal/CC9.1.pdf r is the distance the
point is from the origin (0,0). http://archives.math.utk.edu/visual.calculus/0/polar.6/ Rectangular coordinates, (x,y), represent how far over and how far up you go from the origin (0,0). Polar coordinates, (r,θ), represent how far away from the origin and what angle from the horizontal axis. How are the values of
x, r, y and θ related? Which coordinate system should we use? Graphing in polar coordinates Types of graphs Derivatives of polar equations Let's practice graphing... Areas/Integrals in polar coordinates Spherical Coordinates? Converting from Rectangular to Polar Converting from Polar to Rectangular Rectangular Coordinate (x,y) Polar Coordinate (r,θ) These equations for converting between the two coordinates systems can be found by solving the triangle on the left that represents the point (x,y) or (r,θ) Let's get some practice converting between the systems. The first two we will work through together and the second two you will do on your own. 1) 2) 3) 4) Convert (-1,-2) to a polar coordinate Convert (2,4) to a polar coordinate Convert (3,60°) to a rectangular coordinate Convert (1,145°) to a rectangular coordinate There is one issue
with these equations.
point is in, the inverse tangent function can give us an incorrect value. www.mathsisfun.com/polar-cartesian-coordinates.html The figure on the left shows the
four quadrants of the plane. When
converting from rectangular to
polar coordinates, refer to this to