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### Converting and Graphing Coordinates

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Date: 5/28/96 at 21:25:26
From: Anonymous
Subject:Converting Between Rectangular and Polar Coordinates

I want to know how to convert decimals to polar coordinates, and vice
versa. Also, how would you graph it on a x,y coordinate?
For example, graphing (-1, 3pi/6).

Nanci Echon
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```
Date: 6/3/96 at 11:2:38
From: Doctor Brian
Subject: Re: Converting Between Rectangular and Polar Coordinates

There are two formulas for converting between rectangular coordinates
(x,y) and polar coordinates (r,theta).  They are:

1a) r = sqrt(x^2 + y^2)
1b) theta = arctan(y/x)

2a) x = r cos theta
2b) y = r sin theta

What does this mean?  Well, the radius is thought of as the distance
from the point (x,y) to the origin (0,0). The equation (1a) then comes
from either the distance formula or the Pythagorean theorem.  The
angle (1b) measures displacement of the ray between the point and the
origin with the positive x-axis. (For example, the point (2,2) has an
angle of 45 degrees, and the point (-2,2) has an angle of 135 degrees)

The reverse formula (2a and 2b) comes from the right triangle you can
draw with the origin and x,y with the x-axis as one of the legs.
Using the formulas for sine and cosine you can isolate to solve for x
and y.

Now, the point (-1, 3pi/6) means a radius of -1.  You usually don't
see a negative radius, but what it means is that we will go 1 unit in
the OPPOSITE direction. The direction is determined by 3pi/6 = 90
degrees. But since r=-1, we want to go 1 unit in the direction of 270
degrees instead (the opposite direction). We can either plug in
(1,270) into the above formulas, or just realize that we will end up 1
unit straight down from the origin, at (0,-1).

-Doctor Brian,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
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Associated Topics:
High School Equations, Graphs, Translations

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