I am trying to solve an equation
135 degrees = -(arctan(f/(10^5) + arctan(f/(3.16*10^5))+arctan(f/(10^6)) .
When I enter this equation using the solve function on the Ti-89, it gives a single solution, but I believe there should be 3 solutions. It gives me a warning saying more solutions may exist.
How do I solve this equation using the TI-89 to get all 3 solutions?
There is, in fact, only one solution. By taking the tangents of both sides you do indeed get a cubic equation for $f$ (or for $g=10^{-5}f$, which is easier to work with), and it has three real roots. But then when you plug each root into the original equation you discover that only one solution matches that equation; the others give a discrepancy of $\pi$, which is the period of the tangent function you applied to get the polynomial equation.