In the two Sketches below you can explore the Trigonometric Sine and Cosine.
In the left Sketch both are expressed with respect to the variable t,
while the second Sketch illustrates how they constitute ellipses parametrically.
The four Control Bars below affect both Sketches.
The variable t in the left Sketch is dragable and
the corresponding points on curves are shown in both Sketches.
Graphs of Sine and Cosine
y = r cos at y = s sin bt
The parametric curve in the Phase Plane
ì í
î
x
=
r cos at
y
=
s sin bt
Problems: 1. Keep r = a = s = 1.
Find a value b so that the curve on the right look like a circle.
2. How does the parametric curve behave when the sign of r is changed negative?
3. Keep fixed a = s = b = 1 and see what happens when r goes to zero.
4. What if you keep r = s = 1 and change a and b so that they become equal again
(for example, both up to 3 or 4)?
5. How many times does the parametric curve intersect itself, when a = 4 and b = 3?
6. When you keep a = 1 fixed and put b = 3, 4, 5, what do you get in the Phase Plane?