To solve this problem, let's analyze the triangle formed by points $A$, $B$, and $C$, where:

• Point $A$ and $B$ are located along the shoreline.
• $C$ is the boat's location in the water.
• $\angle A = 42.3^\circ$
• $\angle B = 68.9^\circ$
• $AB = 120$ miles (distance between points $A$ and $B$ along the shoreline).

Since we know two angles in the triangle ($\angle A$ and $\angle B$), we can determine the third angle, $\angle C$, by using the property that the sum of the angles in a triangle is $180^\circ$.

Step 1: Calculate $\angle C$

$\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 42.3^\circ - 68.9^\circ = 68.8^\circ$

Step 2: Apply the Law of Sines

We need to find the distance from point $C$ (the boat) to the shoreline, which is the length $AC$ or $BC$. Using the Law of Sines, we have: $\frac{AB}{\sin(\angle C)} = \frac{AC}{\sin(\angle B)} = \frac{BC}{\sin(\angle A)}$

Let's solve for $AC$: $AC = \frac{AB \cdot \sin(\angle B)}{\sin(\angle C)}$

Step 3: Substitute Known Values

Substitute $AB = 120$ miles, $\angle B = 68.9^\circ$, and $\angle C = 68.8^\circ$: $AC = \frac{120 \cdot \sin(68.9^\circ)}{\sin(68.8^\circ)}$

Using approximate values for the sines: $\sin(68.9^\circ) \approx 0.9272 \quad \text{and} \quad \sin(68.8^\circ) \approx 0.9270$

$AC \approx \frac{120 \cdot 0.9272}{0.9270} \approx 120.03 \text{ miles}$

Conclusion

The distance from the boat to the shoreline (along line $AC$) is approximately 120.03 miles.

Would you like a more detailed breakdown or have any questions?

Here are 5 related questions to deepen your understanding:

1. How would the solution change if the distance $AB$ were different?
2. Can the Law of Cosines be used in this scenario? If so, how?
3. What other applications does the Law of Sines have in navigation?
4. How does this problem illustrate the importance of angle measurements in trigonometry?
5. What if the angle measurements were in radians instead of degrees? How would that affect the solution?

Tip: When solving trigonometric problems involving distances and angles, always double-check angle sum calculations to avoid errors.