Varying Coefficients & Powers of Cosine and Sine

In this write-up I will explore some of the results from varying coefficients and powers of certain parametric equations. In my view, these investigations provide some lovely pictures and some interesting mathematics!

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

x = a cos(t)

y = b sin(t)

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

x = cos(at)

y = sin(bt)

**III.**** Varying powers
cosine and sine in the equations**

x = a cos(t)

y = b sin(t)

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

x = a cos(t)

y = b sin(t)

In this situation, vary the coefficients of the trigonometric
functions themselves. This exploration may be familiar to you
already. If ** a** =

In the graphs above,

The graphs will be circles centered at the origin with radius
** a**. Why? Notice that x^2 + y^2 = a^2(cos(t))^2 +
a^2(sin(t))^2 = a^2[(cos(t))^2 + (sin(t))^2] = a^2. And x^2 +
y^2 = a^2 is the equation of a circle of radius

If ** a** >

In the graphs above,

The graphs will be ellipses centered at the origin with horizontal
major axis **2 a** and vertical minor axis

And, for ** a** >

Of course, when ** a** <

In the graphs above,

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

x = cos(at)

y = sin(bt)

Varying the coefficients of ** t** produces more intriguing
results than those above (in this writer's opinion.) Below are
some graphs where

Of course, it would be natural to investigate whether the curves
above ever "close," since when ** a** = 1,
for instance, the graph is a closed curve (a circle; see above.)
So, let

Notice that the curves do close for ** a** = 0.5 and

To explore further, **download this
Graphing Calculator File**.

- You can play the animation by clicking ONCE on the arrow
button next to the "n" below the graph. Remember that
the values of n are the values of
, the coefficient of*a*in the original equation for x.*t*

- The slider values of n are set to run from 0 to 10 in 100
steps, but you can change these values by double-clicking on
n. You can also change the range of
and the values of*t*by clicking on the values and typing.*b*

- Finally, there is an option to change the coefficients of
cosine and sine (called
and*c*) in the file, which will stretch or compress the graph as noted in part I. above.*d*

Now consider varying ** b** and letting

Below are the graphs where ** b** varies from 0.5
to 1.5 to 2 and

Certainly they do not match the graphs above, but interestingly,
the graph when ** b** = 2 (and

When the range of ** t** is expanded to run from 0
to 4p, notice that the graph when

Now consider the graphs where ** b** is positive integer
(and

It appears that the number of loops in the graph is the same
as the value of ** b**. In addition, as the value of

To see this "filling up" and to explore further,
**download this Graphing Calculator File**.
This time, the values of the slider variable n are the values
of ** b**. (See instructions above as necessary.)

**III. Varying
powers of cosine and sine in the equations**

x = a cos(t)

y = b sin(t)

Building off of part I., here's an exploration of what happens to the graphs of

x = a (cos(t))^n

y = a (sin(t))^nfrom

= 0 to 2p.tNote that if

does not equalb, then the graph will be stretched or compressed in the horizontal and/or vertical directions.a

Below is a graph of ** a** =

Notice that even powers seem to produce graphs only in the
first quadrant, while odd powers produce a graph in four quadrants.
(This observation is true even if ** t** is expanded
to run from 0 to larger multiplies of p.)

To explore further, **this Graphing
Calculator file** will produce the figure above and allow
you to change values for ** a**,

To produce a graph with an even power (even value of n) which
repeats in all four quadrants, reflections can be used. In the
graph below, four sets of equations have been used, with ** t**
running from 0 to 2p. For the red graph,

**This Graphing Calculator file**
will allow you to watch the graph of

x = a (cos(t))^n

y = a (sin(t))^n

change when the power n is a sliding variable; you will see that the graph exists in all four quadrants only at the odd integers (i.e., for most values of n, the graph appears to exist only in the first quadrant.) Remember to click ONCE on the arrow button next to the "n" to start the animation.