# Using HomotopyContinuation.jl to find the roots of systems that include sin and cosine functions?

I am new to Homotopy continuation methods, but I understand the basics. While HC was designed for finding the roots of systems of polynomial equations, I was wondering in the method can be used to find the roots of a system of polynomial equations that includes sin, cosine or other trigonometric functions? I was looking in the docs for the package, but I did not find any reference to this.

So if I have a system such as a simple inverted pendulum, the equations look like the following system of [x_1, x_2]. I set the parameters g=9.8, L = 1.0.

x_1 = x_2\\ x_2 = -\frac{g}{L}sin{x_1}

The solutions to this system are [0, \pm \pi]. I wanted to see if I could use HC to find the roots of a system like this, but I was not sure of how to code this? Do I need to reparameterize the sin function into a polynomial?

Here is some code I was starting with, but it is not working:

    using HomotopyContinuation
@var x y;
x1 = y
x2 = -9.8*x
x3 = x^2 + y^2 - 1
H = System([x1, x2, x3])
result3 = HomotopyContinuation.solve(H)


Any suggestions would be appreciated. Thanks.