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From the image, I can see a trigonometry problem involving bearings and distances between ships and a point. The problem consists of the following parts:

1. **Two ships leave point \( P \) at the same time.**
   - Ship \( A \) sails on a bearing of \( 120^\circ \) from point \( P \).
   - Ship \( B \) sails on a bearing of \( 045^\circ \) from point \( P \).

   After 1 hour:
   - Ship \( A \) is 150 km away from \( P \).
   - Ship \( B \) is 180 km away from \( P \).

2. The diagram is a triangle showing the positions of \( P \), \( A \), and \( B \), forming a triangle \( \triangle PAB \).

### Questions:

**(a)** Find the bearing of \( A \) from \( B \).

**(b)** Find the distance between the two ships \( A \) and \( B \).

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Let's start by solving part **(b)** using the Law of Cosines.

### Step 1: Calculate the angle between the two paths of the ships
- The bearing of ship \( A \) is \( 120^\circ \), and the bearing of ship \( B \) is \( 045^\circ \).
- The angle between these two bearings can be calculated as \( 120^\circ - 45^\circ = 75^\circ \).

This angle is the internal angle \( \angle APB \) between the two ships at point \( P \).

### Step 2: Apply the Law of Cosines to find the distance between ships \( A \) and \( B \)

The Law of Cosines states:
\[
AB^2 = PA^2 + PB^2 - 2 \cdot PA \cdot PB \cdot \cos(\angle APB)
\]
Where:
- \( PA = 150 \) km (distance from \( P \) to ship \( A \)),
- \( PB = 180 \) km (distance from \( P \) to ship \( B \)),
- \( \angle APB = 75^\circ \) (calculated from the bearings).

Substitute the values into the formula:
\[
AB^2 = 150^2 + 180^2 - 2 \cdot 150 \cdot 180 \cdot \cos(75^\circ)
\]

Let me calculate that next.The distance between ships \( A \) and \( B \) is approximately \( 202.3 \) km.

Would you like to proceed with part **(a)** or have any questions on this? 

Here are 5 related questions to deepen your understanding:
1. How is the Law of Cosines useful in real-world navigation?
2. Why do we need to calculate the angle between the ships' bearings before using the Law of Cosines?
3. What other situations might involve solving for distances using bearings and angles?
4. Can the Law of Sines be used in a similar scenario? When would that be appropriate?
5. How would the calculation change if the ships traveled for different amounts of time?

**Tip:** Always remember to convert angles from degrees to radians when using trigonometric functions in most calculators or programming languages.

Two ships leave Point P at the same time. Ship A sails on a bearing of 120° and Ship B sails on a bearing of 045°. After 1 hour, Ship A is 150 km from P and Ship B is 180 km from P. Find the distance between the two ships and the bearing of A from B.

This problem involves two ships leaving point P at different bearings, requiring the use of the Law of Cosines to find the distance between them. Learn how to calculate distances and bearings with a step-by-step solution, including trigonometric concepts and formulas. Ideal for high school students in Grades 9-12.

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