**The objective of this writeup
is to explore results from varying coefficients and power of certain
parametric equations. I will investigate what will happen to by
:**

**1. Varying values of a and
b in the equations**

**11. Varying values of a
and b in the equations**

**1. varying values of a and
b in the equation**

**Let us try to vary the coefficients
a and b in the trigonometric equation. If
we let a = b and t ranges from 0 to 2p**

**The graph above a=b =1;
a=b =1.5; a=b =0.7 for t from 0 to 2p**

**One should note that the
graphs are circles centered at the origin with radius a.
This is so because:**

**x^2 + y^2= a^2(cos(t))^2
+a^2(sin(t))^2 =a^2[(cos(t))^2 +(sin(t))^2] =a^2 +y^2 =a^2 is
the equation of circle of radius a and centered
at the origin.**

**Now suppose a > b, and
t runs from 0 to 2p
suppose we let 1. a = 1.5
and b = 1, 11. a =2 and b =1.5, 111.a =2.5 and b = 2**

**the graph will be an ellipse
centered at the origin as shown above. Let us investigate why?
notice x^2 = a^2 (cos(t))^2 and y^2 = b^2(sin(t))^2, therefore
x^2/a^2 + y^2/b^2 = cos(t)^2 + sin(t)^2 = 1 Ok! Therefore for
a > b in this case is an equation of ellipse with the major
axis along the x - axis and center (o, o)**

**What do you think will happen
when a<b**

**try 1. a = 1 and b = 1.5
11. a = 1.5 and b = 2 111. a = 2 and b = 2.5**

**Case 11. Varying value of a and b in the
equation of**

**x = cos(at)**

**y = sin(bt)**

**Notice the position of a and b in this case.
Let us vary a from 0.5 to 1.5 to 2 but keep b constant at b =1
while t ranges from 0 to 2p**

Now suppose we let t ranges from 0 to 2p the following
graphs will be obtained.

**Notice that the curves do close for a =0.5
and a = 1.5, but for a = 2 ; the graph appear to be oscillating
along the curve for a = 2**

**for more exploration here
for graphing calculator animation**

**Let us vary b i.e b = 0.5, 1.5 and 2 , but
let a = 1**

**One can note that the two graphs (i.e one
keeping a constant and keeping b constant) do not match . Click
here fore graphing
calculator for animation to see what I'm discussing.**

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