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!