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Polar Coordinates From Cartesian Coordinates

Date: 07/27/98 at 02:48:46
From: cherry
Subject: Polar coordinates

Find a pair of polar coordinates of the point whose cartesian 
coordinates are (3, -3 sqrt(3)). 

Please help me answer this.

Date: 07/27/98 at 17:23:55
From: Doctor Jaffee
Subject: Re: Polar coordinates

Hi Cherry,

The easiest way I know how to solve a problem like this is to start by 
drawing a picture. Try it. First draw a coordinate grid and locate the 
point (3,-3 sqrt(3)). Then draw a segment that connects that point to 
the origin. Draw another segment from the point perpendicular to the 

What you should see now is a right triangle whose legs have lengths 3 
and 3 sqrt(3). Using the Pythagorean Theorem you should be able to 
calculate the length of the hypotenuse. You should get 6.

Now, do you remember the theorem from geometry that says that any 
time the leg of a right triangle is half the hypotenuse, you have a 
30-60-right triangle? That means that the angle between the positive 
x-axis and the hypotenuse is 60 degrees.

In order to put the point into polar coordinates, we need to know two 
things: the distance from the point to the origin (we figured out that 
it was 6), and the angle that the segment from the origin to the point 
makes with the positive x-axis (we figured out that it was 60 degrees 
in a clockwise direction so we'll call it -60 degrees).

So, an answer to your question is (6, -60).

That is not the only answer, of course. Instead of calling the angle 
-60, we could have called it 300 degrees. (-6, 120) is another answer.  
There are infinitely many of them.

I hope this explanation helps. Write back if you need more help with 
the problem or when you have another question.

- Doctor Jaffee, The Math Forum
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Associated Topics:
High School Equations, Graphs, Translations
High School Imaginary/Complex Numbers

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