Extra Explorations

By

Brandon Samples

Let's begin by exploring the equation

(sin(x^2))^2 + (sin(y^2))^2 = 1.

Now, can we raise the constant above 1? Well, if we think about the trig functions sine and cosine, we remember that the functions are trapped between -1 and 1 so we must keep the constant below 1. What will happen when we let the constant change?

How about leting the constant equal 1/2?

So it appears that the lines are getting further apart. What if we had set the constant equal to 3/4? Would we expect them to be in between.

Now, what should we expect when we let the constant tend to zero?

Let's look at an animation.