



In algebra, and up to now in trig,




graphs have had an axis running left to right 


This axis is
generally labeled X. 


There is also
an axis running up and down. 


This axis is
usually labeled Y. 





There is
another way to draw a graph. 


This way is
called a polar graph. 


It is set up
to look something like a wheel. 


It has a
center point. 


It has lines
going out from the center point like spokes of a wheel. 


And it has
rings around the center point. 


The whole
thing actually winds up looking like a dart board. 





The center
point is labeled zero. 


The rings are
numbered by their distance from the center. 


In the picture
above, the rings might be numbered 1,2,3 and 4 


going out from the
center. 








Realize
that there is nothing magic about the numbers 1,2,3,and 4. 


The rings
might also be numbered 5,10,15, and 20. 





The spokes are
labeled with degrees, 


starting with the spoke at 3 O'clock and going
counter clockwise. 











In an X Y
graph, points are labeled with two numbers, 


one for the X axis and
one for the Y axis. 





In a polar
graph, a point is also labeled with two numbers. 


The first
number is the distance the point is from the center. 


The second number is the degree line the point is on. 





So
the point (2,30°)
would be: 








There
is nothing really better or worse about polar graphing in general. 


There
are just times when it works better than other types of graphs. 


Those
times are usually when you are graphing equations that have trig
functions. 





One
trick you should know is how to convert something plotted on an X Y
graph 


to a polar graph. 


and
back again. 





Let's
look at the point (3,2) on an X Y graph. 


I
bet you have made right triangles on X Y graphs 


to find the distance
to a point before. 


We'll
do that again here. 








The
point at the end of the hypotenuse (3,2) is our point. 


The
question is, how far is it from the origin (0,0) to our point (3,2). 





To
find the answer, just use the Pythagorean theorem: 





h^{2}
= 2^{2} + 3^{2} 


h^{2}
= 4 + 9 


h^{2}
= 13 


h = 


h =
3.606 





What is the
measure of the little angle at the center of the graph? (Call it A) 


The
tangent of an angle is equal to the length of the side opposite 


divided by the length of the side adjacent. 


So: 




















So
for the X Y graph point: (3,2) the polar coordinates are: (3.606, 33.69°) 





Plotting
that on the polar graph, we get: 








Now
let's go from polar coordinates to X Y (called rectangular)
coordinates. 





Find
the X Y (rectangular) coordinates for the polar point (5,60°) 











We
draw a right triangle on an X Y graph 


with a hypotenuse of 5 and a
60°
angle on the left. 











Now
we need to solve the triangle for X and Y. 











Y =
5 sin 60° 











X = 5 cos 60° 





Let's
stop right here for a moment. 


We
wanted to find the X and Y coordinates of the polar point. 





(5, 60°) 


And
we found: 


X = 5 cos 60° 


and 


Y =
5 sin 60° 





In general, for a polar point (a, B°) the X and Y coordinates are: 





X = a cos B° 


Y = a sin B° 





So let's solve it already! 





X = 5 cos 60° 

 X = 5 (0.5) 


X = 2.5 





Y = 5 sin 60° 


Y = 5 (0.8660) 

 Y = 4.33 





So
the X Y coordinates of the polar point (5,60°) are (2.5,4.33) 








copyright 2008 Bruce Kirkpatrick

