We are given the equation:

$t = \frac{x}{b} + \frac{2z \cos(\theta)}{v}$

We need to solve for $z$. Follow these steps:

Step 1: Subtract $\frac{x}{b}$ from both sides.

$t - \frac{x}{b} = \frac{2z \cos(\theta)}{v}$

Step 2: Multiply both sides of the equation by $v$ to eliminate the denominator on the right-hand side.

$v \left( t - \frac{x}{b} \right) = 2z \cos(\theta)$

Step 3: Divide both sides by $2 \cos(\theta)$ to solve for $z$.

$z = \frac{v \left( t - \frac{x}{b} \right)}{2 \cos(\theta)}$

Thus, the solution for $z$ is:

$z = \frac{v \left( t - \frac{x}{b} \right)}{2 \cos(\theta)}$

Would you like more details on the steps, or do you have any questions?

Here are 5 relative questions to expand on this:

1. How can we apply this formula in a real-world scenario?
2. What happens to $z$ when $\theta = 90^\circ$?
3. How does changing $v$ affect the value of $z$?
4. Can you explain why $\cos(\theta)$ appears in the equation?
5. How would you solve for $\theta$ if $z$ is known?

Tip: In trigonometric equations, make sure to check whether certain angles (like $90^\circ$ or $180^\circ$) make terms like $\cos(\theta)$ zero, which can lead to undefined results.