**Assignment 11: A Look at Polar Equations**

**By Rebecca L. Adcock**

# Polar equations have nothing to do with big
white bears that live in cold places. So what are they? (I had to look this up
in a textbook; polar equations were invented after I earned my BA.)

We just finished looking at
parametric equations in Assignment 10 and polar equations are another way of
looking at parametric equations (or vice versa).

According to __Algebra and
Trigonometry __by James Stewart, Lothar Redlin, and Saleem Watson (page 803)É.

**POLAR EQUATIONS IN PARAMETRIC FORM **

__ __

**The graph of the polar equation r=f(theta) is the same
as the graph of the parametric equations**

** x=f(t)
cos t y=f(t) sin t**

LetÕs take a look at

###

When n is an integer, this
form is called the n-leaf rose.

We can very quickly see a
pattern here.

is a n-leaved rose if n is
odd or a 2n-leaved rose if n is even.

What
happens if we replace cos with sin?

Our
earlier conclusion on the relation of number of leaves to the coefficient of
theta holds true. The orientation of the figure to the axes changed when we
changed cosine to sine

LetÕs see what happens with
the following equation when we vary a,and b .

**a>b** This shows a 3-leaf rose where b= 1 and a varies
from 1 to 3 in increments of .25.

**a<b** This shows a 3-leaf rose where a=1 and b varies from
1.25 to 2 in increments of .25.

** **

**a=b **This shows a 3-leaf rose where a=b and the value
ranges from 1 to 2 in increments of .50.

** **

We have seen that increasing the value of a over b
results in decreasing the definition of our figure. The leaves appear to
Ôsmooth outÕ. Increasing b over a gives our figure more definition.

WeÕve only seen a sample of polar equations. Let me leave
you with one more interesting figure. Looks what happens to the 1-leaf when a=1
and b is greater. Would the figure eventually become a circle?

See yaÉ

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