Eliminate the parameter of parametric curves?

Question

Eliminate the parameter to find a Cartesian equation of the curve:

x=sine of theta, y=cosine of theta, 0 (>or=) theta (>or=) pi

The solution I found was:

y=cos(sin^-1(x))

however, the solution in the book is:

x^2 + y^2 = 1

I don't understand why the answer isn't in "y=" form like every other problem in this section until now. Also, I don't know how to break down the equation past the inverse trig functions. I appreciate any help.

Answer 1

First way:

x = sin(theta) and y = cos(theta)

x^2 = sin^2(theta) and y^2 = cos^2(theta)

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

Second way (although not officially correct - see my note at the bottom):

y = cos(sin^-1(x))

y^2 = cos^2(sin^-1(x)) = 1 - sin^2(sin^-1(x))

but sin(sin^-1(x)) = x, so we can rewrite this as:

y^2 = 1 - x^2

which is the same as the book's solution.

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The reason it's not in "y=" form is that officially, theta is not equal to sin^-1(x).  Why not, you might ask? Well, what is the range of the _function_ sin^-1?