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Solve find the missing side or angle show set-up ratio and solution

To solve for the missing side \( x \) in the given right triangle, we can use trigonometric ratios. 

### Given:
- The angle is \( 42^\circ \).
- The side opposite the angle (opposite side) is \( 10 \) meters.
- The side \( x \) is adjacent to the angle \( 42^\circ \).

### Solution:

Since we know the angle and the length of the opposite side, we use the tangent function, which relates the opposite side to the adjacent side:

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{10}{x}
\]

Here, \( \theta = 42^\circ \), so:

\[
\tan(42^\circ) = \frac{10}{x}
\]

Now, solve for \( x \):

\[
x = \frac{10}{\tan(42^\circ)}
\]

Let's calculate it:

\[
x \approx \frac{10}{0.9004} \approx 11.11 \text{ meters}
\]

### Final Answer:
The length of side \( x \) is approximately \( 11.11 \) meters.

Would you like more details or have any questions about this problem?

Here are five related questions to explore further:
1. How would you find the hypotenuse of this triangle?
2. What would be the angle \( \theta \) if side \( x \) were given as \( 15 \) meters?
3. How do you find the area of this triangle using the given information?
4. Can you solve for the missing side using the sine or cosine functions instead?
5. What would be the effect on \( x \) if the angle \( 42^\circ \) were increased to \( 60^\circ \)?

**Tip:** When dealing with right triangles, it's helpful to know the primary trigonometric ratios (sine, cosine, tangent) and when to apply each based on the given information.

Learn how to find the missing side \( x \) in a right triangle with angle 42 degrees using trigonometric ratios. Detailed step-by-step solution with calculations. Suitable for high school students studying trigonometry.

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