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)


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.