We aim to solve the system of equations for $a$ and $b$ in terms of $r$ and $\theta$ explicitly.

From the image, the equations are:

1. $(c - r \cos\theta)^2 \big(a(c - r \cos\theta) + b\big) = r \sin\theta$

2. $(c - r \cos\theta)\big(3a - (c - r \cos\theta) + 2b\big) = \frac{\cos\theta}{\sin\theta}$

We will proceed to isolate $a$ and $b$. Let me compute this step-by-step. sympy import symbols, Eq, solve, cos, sin

Define variables

a, b, c, r, theta = symbols('a b c r theta')

Define the two equations

eq1 = Eq((c - rcos(theta))2 * (a(c - rcos(theta)) + b), rsin(theta)) eq2 = Eq((c - rcos(theta)) * (3a - (c - rcos(theta)) + 2b), cos(theta)/sin(theta))

Solve the system of equations for a and b

solution = solve([eq1, eq2], (a, b)) solutionThe explicit solutions for $a$ and $b$ in terms of $r$, $\theta$, and $c$ are:

Solution for $a$:

$a = \frac{c^3 \sin\theta - 3 c^2 r \cos\theta \sin\theta + 3 c r^2 \cos^2\theta \sin\theta + c \cos\theta - r^3 \cos^3\theta \sin\theta - 2 r \sin^2\theta - r \cos^2\theta}{-2 c^3 \sin\theta + 6 c^2 r \cos\theta \sin\theta + 3 c^2 \sin\theta - 6 c r^2 \cos^2\theta \sin\theta - 6 c r \cos\theta \sin\theta + 2 r^3 \cos^3\theta \sin\theta + 3 r^2 \cos^2\theta \sin\theta}$

Solution for $b$:

$b = \frac{-c^4 \sin\theta + 4 c^3 r \cos\theta \sin\theta - 6 c^2 r^2 \cos^2\theta \sin\theta - c^2 \cos\theta + 4 c r^3 \cos^3\theta \sin\theta + 2 c r \cos^2\theta - r^4 \cos^4\theta \sin\theta - r^2 \cos^3\theta + 3 r \sin^2\theta}{-2 c^3 \sin\theta + 6 c^2 r \cos\theta \sin\theta + 3 c^2 \sin\theta - 6 c r^2 \cos^2\theta \sin\theta - 6 c r \cos\theta \sin\theta + 2 r^3 \cos^3\theta \sin\theta + 3 r^2 \cos^2\theta \sin\theta}$

Both $a$ and $b$ are expressed as fractions with the same denominator, which depends on $r$, $\theta$, and $c$.

Next Steps:

Let me know if you would like:

1. A simplification of these expressions for specific cases.
2. Numerical evaluations for specific values of $r$, $\theta$, and $c$.
3. Visualizations or a step-by-step derivation of the solutions.
4. Further insights into the dependency of $a$ and $b$ on $\theta$ and $r$.
5. An exploration of the geometric meaning of these equations.

Tip: When solving trigonometric equations, always check for scenarios where $\sin\theta$ or $\cos\theta$ might lead to undefined results or special cases like $\theta = 0$ or $\pi/2$.