ASSIGNMENT
#10
PARAMETRIC
CURVES

**A
parametric curve in the plane is a pair of functions:**
**
x = f(t)**
**y
= g (t)**
**where
the two continuous funtions define ordered pairs (x,y). The two
equations are usually called the parametric equations of a curve.
The extent of the curve will help depend on the range of ***t.
*In many applications, we think of x and y "varying with
time t" or the angle of rotation that some line makes from
an initial location.

**We
will investigate **
**x
= a cos (t)**
**y
= b sin (t)**
**for
**
**Let's
look at a couple different cases;**
**1.
a=b**
**Let
a = b =2 **
**If
you solve the first equation for cos t and the second equation
for sin t. **
**cos
t = x/2 and sin t = y/2**
**Now
use your trig identity cos^2 t + sin ^2 t =1 to rewrite the equation
to eliminate t. **
**cos
^2 t + sin ^2 t = 1**
**(x/2)^2
+ (y/2)^2 =1 (by subsitution)**
**x^2/4
+y^2/4 =1**
**x^2
+ y^2 = 4 (Multiply each side by 4)**
**As
you can see by looking at the equation or the graph this is a
circle with center at (0,0) and a radius of 2. As t increases
from **
**2.
a < b**
**Let
a =1 and b =2 **
**Notice
that b is larger than a, since sin is connected with the y- coornidate,
the graph did exactly what should be expected it should strech
vertically. Now can you determine what will happen if we make
b smaller than a?**
**3.
a > b**
**Let
a =2 and b =1 **
**You
probably guessed it. That's right since the x-coorndinate is connected
to cosine then the circle should strech horizontally. **
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** **