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We are tasked with solving a triangle problem using the Law of Sines. The triangle is defined by:

- Points \( A \), \( B \), and \( C \), where \( A \) and \( B \) are on the shoreline, and \( C \) is the sailboat.
- The distance between \( A \) and \( B \) is \( 20 \) meters.
- Angle \( \angle A = 65^\circ \).
- Angle \( \angle B = 42^\circ \).

### Step 1: Find \( \angle C \)

Since the sum of the angles in any triangle is \( 180^\circ \):

\[
\angle C = 180^\circ - \angle A - \angle B = 180^\circ - 65^\circ - 42^\circ = 73^\circ
\]

### Step 2: Apply the Law of Sines

We can use the Law of Sines to find the distances from point \( C \) to \( A \) and \( B \), denoted as \( AC \) and \( BC \).

The Law of Sines states:

\[
\frac{AC}{\sin(\angle B)} = \frac{BC}{\sin(\angle A)} = \frac{AB}{\sin(\angle C)}
\]

Let's first find the distance \( AC \) and \( BC \) using the known side \( AB = 20 \) meters.

### Solving for \( AC \):

\[
\frac{AC}{\sin(42^\circ)} = \frac{20}{\sin(73^\circ)}
\]

Solving this for \( AC \):

\[
AC = \frac{20 \cdot \sin(42^\circ)}{\sin(73^\circ)}
\]

### Solving for \( BC \):

\[
\frac{BC}{\sin(65^\circ)} = \frac{20}{\sin(73^\circ)}
\]

Solving this for \( BC \):

\[
BC = \frac{20 \cdot \sin(65^\circ)}{\sin(73^\circ)}
\]

I will now calculate these values.The distances from the sailboat at point \( C \) to the two observers are:

- Distance from \( C \) to \( A \): \( 13.99 \, \text{meters} \)
- Distance from \( C \) to \( B \): \( 18.95 \, \text{meters} \)

Would you like more details on how the calculations were performed or clarification on any steps?

Here are 5 related questions to explore further:
1. How is the Law of Sines applied to non-right triangles?
2. Can the Law of Cosines be used for this type of problem?
3. What happens if one of the angles is obtuse in a triangle like this?
4. How do changes in the angles affect the distances to the sailboat?
5. Could this problem be solved without trigonometry?

**Tip:** Always ensure that the sum of the angles in a triangle adds up to \( 180^\circ \) to verify if your angle calculations are correct.

Two people at points A and B on the shoreline observe a sailboat at position C. The distance between points A and B on the shoreline is 20m. The first observer measures ∠A = 65° and the second observer measures ∠B = 42°. Find the distance from C to each observer.

Learn how to find the distances from a sailboat to two observers on a shoreline using the Law of Sines. This problem involves applying trigonometric concepts and formulas, suitable for students in Grades 9-12.

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