Can Nonlinear Systems Be Solved Using Gaussian Elimination?

[Saladsamurai]

Homework Statement

This problem shows up in Anton's Elementary Linear Algebra in the first chapter. It's one of the last problems, so I don't think that it is crucial for me to 'solve' it. But I would like to clear up some conceptual questions I have.

First here is the problem statement:

Solve the following system of nonlinear EQs for the unknown angles $\alpha$, $\beta$, and $\gamma$, where

$0\le\alpha\le2\pi$, $0\le\beta\le2\pi$, $0\le\gamma\le\pi$.

$2\sin\alpha - \cos\beta + 3\tan\gamma = 3$
$4\sin\alpha + 2\cos\beta - 2\tan\gamma = 2$
$6\sin\alpha - 3\cos\beta + \tan\gamma = 9$

Here are my questions:

1) In all of this chapter (on elimination methods), we use Gaussian Elimination on systems of linear EQs. Can the elimination methods be used on a nonlinear system?

2) Since tan(gamma) is not defined at pi/2 , well..., I don't know what I am trying to ask.
But surely this restriction will have some sort of impact on the solution(?).

Thoughts?

 

[aPhilosopher]

it's linear in the trigonometric functions so you can just use linear techniques to solve for them. It's invertible too, if I'm not mistaken. So you will get three trig equations to solve. pi/2 is not a valid solution for gamma.

 

[LCKurtz]

Call $$x = sin(\alpha),\ y = \cos(\beta),\ z = \tan(\gamma)$$

and proceed.

 

[Saladsamurai]

Call $$x = sin(\alpha),\ y = \cos(\beta),\ z = \tan(\gamma)$$

and proceed.

Right. This is what i planned on doing, but I just wasn't sure why they were so adamant on letting me know that it is nonlinear.