https://file.aiquickdraw.com/mathbot/file/1725897450472rym1fvo2.jpg

Let's solve the problem step by step:

1. **Identify the sides**:
   - \( a = 8 \) ft (opposite to angle \( \beta \)),
   - \( b = 9 \) ft (adjacent to angle \( \beta \)).

2. **Find the hypotenuse using the Pythagorean Theorem**:
   \[
   c = \sqrt{a^2 + b^2} = \sqrt{8^2 + 9^2} = \sqrt{64 + 81} = \sqrt{145}
   \]
   So, the hypotenuse \( c = \sqrt{145} \) ft.

3. **Trigonometric functions**:
   - \( \sin \beta = \frac{a}{c} = \frac{8}{\sqrt{145}} \),
   - \( \cos \beta = \frac{b}{c} = \frac{9}{\sqrt{145}} \),
   - \( \tan \beta = \frac{a}{b} = \frac{8}{9} \).

4. **Exact values**:
   - \( \sin \beta = \frac{8}{\sqrt{145}} \),
   - \( \cos \beta = \frac{9}{\sqrt{145}} \),
   - \( \tan \beta = \frac{8}{9} \).

5. **Rationalize the denominator for sine and cosine**:
   - \( \sin \beta = \frac{8\sqrt{145}}{145} \),
   - \( \cos \beta = \frac{9\sqrt{145}}{145} \).

6. **Approximate values**:
   Now, let's calculate the decimal approximations.

I'll compute those next.Here are the results:

### Exact values:
- \( \sin \beta = \frac{8}{\sqrt{145}} \approx 0.664 \)
- \( \cos \beta = \frac{9}{\sqrt{145}} \approx 0.747 \)
- \( \tan \beta = \frac{8}{9} = 0.889 \)

### Approximate values:
- \( \sin \beta \approx 0.664 \)
- \( \cos \beta \approx 0.747 \)
- \( \tan \beta \approx 0.889 \)

Would you like more details on any step or clarification?

Here are 5 related questions to explore:
1. How do we rationalize a denominator in a fraction involving square roots?
2. Can you calculate the sine, cosine, and tangent of \( \alpha \) (the other angle)?
3. What is the relationship between sine, cosine, and tangent for complementary angles?
4. How would the trigonometric values change if the triangle's sides were scaled up?
5. Can you verify these trigonometric values using a calculator or graphing tool?

**Tip**: Memorize the exact trigonometric values for common angles (e.g., 30°, 45°, 60°) for quicker problem-solving!

Find sine, cosine, and tangent of \( \beta \), given that \( a = 8 \) ft and \( b = 9 \) ft in a right triangle.

Learn how to find the sine, cosine, and tangent of angle \( \beta \) in a right triangle where the opposite side is 8 ft, the adjacent side is 9 ft, and the hypotenuse is \( \sqrt{145} \). This problem explores key trigonometric concepts, including using the Pythagorean theorem to find the hypotenuse and calculating the trigonometric ratios.

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