parametric curve in the plane is a pair of functions:
x = f(t)
= g (t)
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.
= a cos (t)
= b sin (t)
look at a couple different cases;
a = b =2
you solve the first equation for cos t and the second equation
for sin t.
t = x/2 and sin t = y/2
use your trig identity cos^2 t + sin ^2 t =1 to rewrite the equation
to eliminate t.
^2 t + sin ^2 t = 1
+ (y/2)^2 =1 (by subsitution)
+ y^2 = 4 (Multiply each side by 4)
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
a < b
a =1 and b =2
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?
a > b
a =2 and b =1
probably guessed it. That's right since the x-coorndinate is connected
to cosine then the circle should strech horizontally.
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