This problem involves solving a right triangle, where you are given the following information:

• $a = 3$
• $c = 10$

You need to find:

• $b$ (the length of the other leg),
• $\alpha$ (the angle opposite leg $a$),
• $\beta$ (the angle opposite leg $b$).

Step 1: Use the Pythagorean theorem to find $b$

The Pythagorean theorem states:

$a^2 + b^2 = c^2$

Substitute the known values:

$3^2 + b^2 = 10^2$

$9 + b^2 = 100$

$b^2 = 100 - 9 = 91$

$b = \sqrt{91} \approx 9.53$

So, $b \approx 9.53$.

Step 2: Find $\alpha$ and $\beta$

The angles can be found using trigonometric ratios.

• To find $\alpha$, use the sine or cosine function. We’ll use cosine:

$\cos(\alpha) = \frac{a}{c} = \frac{3}{10} = 0.3$

Now, take the inverse cosine:

$\alpha = \cos^{-1}(0.3) \approx 72.34^\circ$

• To find $\beta$, we know that $\alpha + \beta = 90^\circ$ (since it’s a right triangle):

$\beta = 90^\circ - \alpha \approx 90^\circ - 72.34^\circ \approx 17.66^\circ$

Final answers:

• $b \approx 9.53$
• $\alpha \approx 72.34^\circ$
• $\beta \approx 17.66^\circ$

Would you like more details on any step or clarification? Here's a list of related questions you can explore:

1. How do you use the Pythagorean theorem to solve right triangles?
2. What other trigonometric ratios can be used to find angles in a right triangle?
3. What are the differences between sine, cosine, and tangent in trigonometry?
4. How can you verify your results in trigonometry problems?
5. What are some real-world applications of the Pythagorean theorem?

Tip: Always check if the values you’re solving for make sense, especially in terms of triangle angles, which should add up to 180° (with 90° already being a right angle).