Assignment 10:

Parametric Curves for Powers of Sine and
Cosine

by

Jenny Johnson

**What is a parametric curve?**

A
parametric curve in the plane is a pair of functions x = f(t) and y = g(t) where the two continuous
function define ordered pairs (x, y). The two equations are called the
parametric equations of a curve. The extent of the curve depends on the range
of t.

We
will explore the parametric curves when the parametric equations are different
powers of cos(t) and sin(t).

**What does the parametric curve of x = a cos (t) and y
= b sin (t) look like?**

** **For all of the following explorations, we will keep
the range of t as 0 < t < 2¹. First, let us look at the parametric
curve when a = 1 and b = 1.

The graph is the
unit circle! Now, what would the
parametric look like if we had a =2 and b = 2.

This
parametric curve is a circle centered at the origin with a radius of 2. We could then make the conjecture that
when a = b, the graph of the parametric curve will be a circle centered at the
origin with radius of a (and b).
Below is a movie of the parametric curves when a = b ranges from -5 to
5. Our conjecture that the graphs
will be circles centered at the origin with r = a = b holds true.

Let
us now explore the curve when we keep a = 1 and change the values of b. When b = 2, the curve looks like this:

The
graph is an ellipse. It appears
that the value for a is the distance of the x-intercepts of the parametric
curve from the origin and the value of b is the distance of the y-intercepts of
the parametric curve from the origin.
We can test this conjecture with the following movie with a = 1 and b
values ranging from -5 to 5.

We
see that when b is negative, the y-intercepts are a distance of |b| from the origin. Now,
to test our conjecture of |a| being the distance of the x-intercepts from the
origin, we will view a movie of the curve when b = 1 and the a values range
from -5 to 5.

We can make the
following observations when working with the parametric equations x = a cos (t)
and y = sin (t):

1.
When |a| = |b|, the parametric curve
will be a circle centered at the origin with radius |a| (and also |b|).

2. When
|a| > |b|, the parametric
curve will be an ellipse with the x-axis as the major axis, the x-intercepts a
distance |a| from the origin, and the y-intercepts a distance |b| from the
origin.

3. When |a| < |b|, the
parametric curve is an ellipse with the y-axis as the major axis, the
x-intercepts a distance |a| from the origin, and the y-intercepts a distance
|b| from the origin.

**What does the parametric curve of x = a (cos (t)) ^{2}
and y = b (sin (t))^{2} look like?**

** **When a = 1 and b = 1, we get the following parametric
curve.

** **We will graph this curve on the same axes as the
parametric curve of x = a cos (t) and y = b cos (t) when a = 1 and b = 2.

It appears
this curve will form a segment from the value of a on the x-axis to the value
of b on the y-axis. The following
movie shows both curves while a = 1 and b ranges from -5 to 5.

We
can see that this new curve does form a line segment from the value of a on the
x-axis to the value of b on the y-axis.
We would then expect the line segment to be in Quadrant II when a is
negative and b is positive, and in Quadrant III when a and b are negative. The following picture shows the curves
when a = -3 and b = 2, and when a = -3 and b = -4, respectively.

**What does the parametric curve of x = a (cos (t)) ^{3}
and y = b (sin (t))^{3} look like?**

** **Let a = 1 and b = 1.

From
the graph, we see that the x- and y-intercepts for the parametric curve of
these parametric equations are the same as those for our first parametric
equations. Let us see if this observation
is the same when a = 2 and b = 5.

** **We see that the x-intercepts of this curve are a
distance of |a| from the origin and the y-intercepts are a
distance of |b| from the origin.

**What does the parametric curve of x = a (cos (t)) ^{4}
and y = b (sin (t))^{4} look like?**

** **Starting with a = 1 and b = 1, we get the following graph.

We
see this parametric curve is a curve from the value of a on the x-axis to the value
of b on the y-axis. This is
similar to the curve for x = a (cos (t))^{2} and y = b (sin (t))^{2}
since it starts and ends at the same point, but it is a curve not a segment.

We
can make the prediction that the curve
of x = a (cos (t))^{5} and y = b (cos (t))^{5} will have
x – intercepts of a and –a and y-intercepts of b and –b, but
it will be closer to the pair x- and y-axis than the other curves (since the
curves have come closer to the axes as the exponent for cosine and sine have
increased). Let us see if our
conjecture is true.

We
are right! We can now make a few
observations about the parametric curves based on the exponents for cos (t) and
sin (t) in our parametric curves.
The following conjectures are related to the parametric curves for the
equations x = a (cos (t))^{n} and y = b (sin (t))^{n}. ** **

1.
When n is an odd
number, the parametric curve will have two x-intercepts a and –a and
y-intercepts b and –b.

2.
When n is an even
number, the parametric curve will only be in one quadrant and will be a segment
or curve from (a, 0) to (0, b).

3.
As n increases,
the curve gets closer to the axes.

We
can check our conjectures by graphing curves up to n = 9 on the same set of
axes when a = 4 and b = -2.

All
of our conjectures hold true.