To solve for $x$, which is the length of the hypotenuse in the given right triangle, we use the sine function since we are given the opposite side ($9.9$) and the angle ($19^\circ$).

Steps:

1. Use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Here, $\theta = 19^\circ$, the opposite side is $9.9$, and the hypotenuse is $x$.

2. Substitute the values into the formula: $\sin(19^\circ) = \frac{9.9}{x}$

3. Solve for $x$: $x = \frac{9.9}{\sin(19^\circ)}$

4. Calculate the sine of $19^\circ$: $\sin(19^\circ) \approx 0.32557$

5. Compute $x$: $x = \frac{9.9}{0.32557} \approx 30.4$

Final Answer:

$x \approx 30.4$

Would you like more details or further assistance?

Relative Questions:

1. What are the steps to solve for other sides of a right triangle if given different information?
2. How does the cosine function differ from the sine function in similar problems?
3. Can the tangent function help in solving this triangle?
4. What is the significance of rounding to the nearest tenth in practical applications?
5. How can the Pythagorean theorem be used in solving right triangles?

Tip:

Always double-check your calculations with a calculator to ensure accuracy, especially when working with trigonometric functions!