To solve this problem, we are given a right triangle with the following information:

• Angle $\beta = 32.70^\circ$,
• Side $a = 29.13$ (which represents the side opposite to angle $\beta$).

We need to find the hypotenuse ($c$).

Solution Steps:

We can use the sine function because: $\sin(\beta) = \frac{a}{c}$ Where:

• $\sin(\beta)$ is the sine of angle $\beta$,
• $a = 29.13$ is the opposite side,
• $c$ is the hypotenuse.

Rearranging the equation to solve for $c$: $c = \frac{a}{\sin(\beta)}$

Now substitute the known values: $c = \frac{29.13}{\sin(32.70^\circ)}$

Using a calculator to find $\sin(32.70^\circ)$: $\sin(32.70^\circ) \approx 0.5398$

Now calculate $c$: $c = \frac{29.13}{0.5398} \approx 53.97$

Final Answer:

The length of the hypotenuse is approximately 53.97.

Would you like further details, or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would you calculate the length of the side $b$ adjacent to angle $\beta$?
2. What would change if angle $\beta$ was larger, say $45^\circ$?
3. How do you find angle $\alpha$ in this triangle?
4. What other trigonometric functions could you use to find the hypotenuse?
5. How does the Pythagorean theorem relate to this problem?

Tip: When solving trigonometric problems, always ensure your calculator is in the correct mode (degrees or radians).