A ship sailed 3 km west, then 2 km south.
Give its three-figure bearings from an observer who stayed at its starting point,
correct to the nearest degree.
b For a person on the ship, what would be the three-figure bearings looking back to
the starting point?

### Part (a): Bearings from the Starting Point
1. **Determine the Final Position of the Ship:**
   - The ship first sailed 3 km west, then 2 km south.

2. **Calculate the Angle of the Bearing:**
   - The bearing is measured clockwise from the north direction.
   - The ship's final position is to the southwest of the starting point.

   \[
   \text{tan}(\theta) = \frac{\text{opposite (south)} }{ \text{adjacent (west)} } = \frac{2 \text{ km}}{3 \text{ km}}
   \]

   \[
   \theta = \text{tan}^{-1}\left(\frac{2}{3}\right) \approx 33.69^\circ
   \]

   - This angle is measured from the westward direction toward the south, so the bearing from the north direction is:
  
   \[
   \text{Bearing} = 180^\circ + 33.69^\circ \approx 214^\circ
   \]

   - To the nearest degree, the bearing from the starting point is **214°**.

### Part (b): Bearings from the Ship Back to the Starting Point
1. **Determine the Bearing from the Ship:**
   - To find the bearing looking back to the starting point, you simply add \(180^\circ\) to the bearing found in part (a).

   \[
   \text{Reverse Bearing} = 214^\circ + 180^\circ = 394^\circ
   \]

   - Since bearings are given as three-figure numbers and within \(0^\circ\) to \(360^\circ\), subtract \(360^\circ\) from \(394^\circ\).

   \[
   \text{Reverse Bearing} = 394^\circ - 360^\circ = 34^\circ
   \]

   - So, the bearing looking back to the starting point from the ship is **034°**.

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Do you want more details or have any questions about these calculations? 

Here are 5 related questions to explore further:

1. What is the significance of three-figure bearings in navigation?
2. How would the bearing change if the ship traveled east and then north instead?
3. How does the Pythagorean theorem apply to find the direct distance between the start and end points?
4. What would be the bearing if the ship sailed 5 km west and 5 km north?
5. How are bearings used in aviation compared to marine navigation?

**Tip:** Always remember that bearings are measured clockwise from the north direction.

Learn how to calculate bearings in navigation with this detailed analysis of a ship's path. Understand trigonometric principles applied to determine bearings from both the starting point and the ship's perspective. Suitable for grades 9-12 students studying navigation and trigonometry.

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