I have a nonlinear $3 \times 3$ system of algebraic/trigonometric equations for which I'm trying to prove existence and uniqueness of a solution. Specifically, I'm looking at the system \begin{align} \frac{\sin(\theta-\gamma)}{\sin \gamma} &= \dfrac{u (u + \sqrt{u^2 + 2gr \cos \theta})}{gr} \\ b &= r \frac{2N \sin(2 \gamma) - 2 \sin(2N \gamma) \cos(2\theta + (2N-2) \gamma)}{4 \sin(\theta) \sin(2 \gamma)} \\ a &= r\frac{\sin(2N \gamma) \sin(2 \theta + (2N-2)\gamma)}{2 \sin(\theta) \sin(2\gamma)}. \end{align} which I'm trying to solve for $r, \theta, \gamma$ in terms of given parameters $u\geq 0$, $N \in \mathbb{N}$, $b>0$, and $a \leq 0$.
My first thought is to set up an Implicit Function Theorem argument--maybe fix $N$ and let $F: \mathbb{R}^6 \to \mathbb{R}^3$ be defined by $F(u, a, b, \theta, \gamma, r) = (LHS_1-RHS_1, LHS_2-RHS_2, LHS_3-RHS_3)$ where $LHS_k, RHS_k$ denote the left- and right-hand sides of the $k$th equation. (Somewhat better, perhaps, would be to cross-multiply in each equation so that $F$ only involves multiplication, addition, and subtraction but no division.) I'm kind of intimidated by the Jacobian that results, though. Are there any shortcuts/standard tricks/totally different approaches that might help with this?