This is a right triangle where:

• The hypotenuse is $31$.
• One of the acute angles is $42^\circ$.
• The side opposite the $42^\circ$ angle is labeled as $x$, which needs to be determined.

Step-by-step solution:

To solve for $x$, we use the sine function: $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ Substitute the values: $\sin(42^\circ) = \frac{x}{31}$ Rearranging for $x$: $x = 31 \cdot \sin(42^\circ)$

Calculate:

Using a calculator: $\sin(42^\circ) \approx 0.6691$ $x = 31 \cdot 0.6691 = 20.742$

Final Answer:

$x \approx 20.74$

Would you like me to explain this further or solve for additional unknowns?

Related Questions:

1. How would you find the length of the other leg of the triangle?
2. What is the cosine of the $42^\circ$ angle in this triangle?
3. How does the tangent function relate to this problem?
4. What are the relationships between the angles and sides in a right triangle?
5. How would the solution change if the hypotenuse were longer?

Tip:

For triangles, always verify if a right angle is present before using trigonometric ratios like sine, cosine, or tangent!