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.

This page explains how to solve the equation t = x/b + (2zcos(θ)/v) for z, using a detailed step-by-step approach. It covers key algebraic manipulations and trigonometric functions, including cosine. This solution is ideal for high school students in Grades 10-12 learning algebra and trigonometry.

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