To solve for $x$ in this right triangle using trigonometric ratios, let’s examine the given information:

1. We have a right triangle.
2. The angle opposite the given side is $28^\circ$.
3. The length of the adjacent side (to the angle $28^\circ$) is $9$ meters.
4. We need to find the length of the hypotenuse $x$.

Step-by-Step Solution

Since we know the adjacent side and need to find the hypotenuse, we can use the cosine ratio:

$\cos(28^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}}$

Substitute the known values:

$\cos(28^\circ) = \frac{9}{x}$

Rearrange to solve for $x$:

$x = \frac{9}{\cos(28^\circ)}$

Calculation

Using a calculator:

1. Find $\cos(28^\circ) \approx 0.8829$.
2. Divide $9$ by $0.8829$:

$x \approx \frac{9}{0.8829} \approx 10.19 \text{ meters}$

Answer

The length of $x$ is approximately 10.19 meters, rounded to two decimal places.

Would you like further details on any of these steps or have any questions?

Additional Questions for Practice:

1. How would you find the opposite side in a similar triangle if the hypotenuse were known?
2. What is the value of the sine function for a $28^\circ$ angle?
3. How do you calculate the length of the hypotenuse if given the opposite side instead?
4. Can you use the tangent ratio in this case, and why or why not?
5. How would the answer change if the angle were larger?

Tip:

Always double-check that you are using the correct trigonometric ratio based on the sides given in relation to the specified angle.