To give you a fully expanded solution for $b$ in terms of $r$ and $\theta$, I need more details about the original equation or system involving $b$, $r$, and $\theta$. Without reference to $a$, I will explicitly solve and expand $b$ only in terms of $r$ and $\theta$, assuming an appropriate context.

If you provide me with the original equation or clarify the relationship, I’ll tailor my solution precisely. For now, I’ll construct a generalized example to demonstrate:

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Example Problem:

Suppose we have an equation like this: $b = r^2 \sin(\theta) + r \cos^2(\theta)$

Solution (Expanded in Terms of $r$ and $\theta$):

We will fully expand and simplify the terms.

1. First term: $r^2 \sin(\theta)$

2. Second term: Expand $\cos^2(\theta)$ using the Pythagorean identity: $\cos^2(\theta) = 1 - \sin^2(\theta)$ Thus, the second term becomes: $r \cos^2(\theta) = r (1 - \sin^2(\theta)) = r - r \sin^2(\theta)$

3. Combine terms: Substitute the expanded second term back into the equation for $b$: $b = r^2 \sin(\theta) + r - r \sin^2(\theta)$

4. Fully expanded result: Combine like terms to give the final expression: $b = r^2 \sin(\theta) - r \sin^2(\theta) + r$

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If this doesn't match your problem, feel free to upload the equation or provide more context so I can give you the exact solution.

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Follow-up Questions:

1. Can you clarify the original equation or provide additional details?
2. Does the expanded solution meet your needs, or should I rewrite it differently?
3. Is there a specific trigonometric identity or simplification you’re looking for?
4. Are there restrictions or specific ranges for $r$ or $\theta$?
5. Would you like help visualizing this solution (e.g., a plot)?

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Tip:

When solving equations with trigonometric components, always check for possible simplifications using identities like $\sin^2(\theta) + \cos^2(\theta) = 1$. These can often make your results more compact.