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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).

The right triangle below has β = 32.70° and a = 29.13. Give the length of the hypotenuse with two decimals and without a unit (e.g. 35.00).

Learn how to solve for the hypotenuse of a right triangle with β = 32.70° and a = 29.13 using the sine function. This problem covers key trigonometric concepts and formulas for calculating the hypotenuse. Suitable for students in Grades 9-10.

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