The problem involves a right triangle with:

• $a = 33.75$ (hypotenuse)
• $\alpha = 68^\circ$ (angle at vertex $A$)

We are tasked with calculating $b$ (the side opposite angle $\alpha$) and rounding it to 1 decimal place.

Step-by-Step Solution:

In a right triangle, the side opposite an angle can be calculated using the sine function:

$\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}$

Here: $\sin(68^\circ) = \frac{b}{33.75}$

Rearranging for $b$: $b = 33.75 \cdot \sin(68^\circ)$

Calculation:

Using a calculator:

1. $\sin(68^\circ) \approx 0.9272$
2. $b = 33.75 \cdot 0.9272$
3. $b \approx 31.3$ (rounded to 1 decimal place)

Final Answer:

$b = 31.3$

Would you like me to calculate other sides or verify? Let me know!

Related Questions:

1. How is the sine function defined in a right triangle?
2. Can you calculate $c$ (adjacent side to $\alpha$) in this triangle?
3. What is the value of $\beta$ (the angle at $B$)?
4. What is the perimeter of the triangle?
5. How can trigonometric identities help solve similar problems?

Tip:

Always ensure your calculator is set to degrees (not radians) when working with angles like $68^\circ$.