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The diagram shows the location of three hospitals. Hospital B is 9 kilometres from hospital A on a
bearing of 217◦. Hospital C is 15 kilometres from hospital A on a bearing of 130◦. What is the bearing of hospital C from hospital B?

To determine the bearing of Hospital C from Hospital B, we'll use trigonometry and vector addition in the context of bearings and distances.

First, let's summarize the given information:
- Hospital B is 9 kilometers from Hospital A on a bearing of 217°.
- Hospital C is 15 kilometers from Hospital A on a bearing of 130°.

We need to find the bearing of Hospital C from Hospital B.

1. **Convert the bearings into Cartesian coordinates:**
   - For Hospital B from Hospital A (bearing 217°):
     - This is measured clockwise from the north direction.
     - Convert the bearing to an angle in standard position (measured counterclockwise from the positive x-axis): \( 360° - 217° = 143° \).
     - Using trigonometric functions:
       \[
       B_x = 9 \cos(143°)
       \]
       \[
       B_y = 9 \sin(143°)
       \]

   - For Hospital C from Hospital A (bearing 130°):
     - Convert the bearing to an angle in standard position: \( 360° - 130° = 230° \) (measured counterclockwise from the positive x-axis, this is simply 130° directly)
     - Using trigonometric functions:
       \[
       C_x = 15 \cos(130°)
       \]
       \[
       C_y = 15 \sin(130°)
       \]

2. **Calculate the coordinates of Hospitals B and C relative to Hospital A:**
   - For Hospital B:
     \[
     B_x = 9 \cos(143°) = 9 \cos(143°)
     \]
     \[
     B_y = 9 \sin(143°) = 9 \sin(143°)
     \]

   - For Hospital C:
     \[
     C_x = 15 \cos(130°) = 15 \cos(130°)
     \]
     \[
     C_y = 15 \sin(130°) = 15 \sin(130°)
     \]

3. **Calculate the position of Hospital C relative to Hospital B:**
   - The coordinates of C relative to B are:
     \[
     C_{Bx} = C_x - B_x
     \]
     \[
     C_{By} = C_y - B_y
     \]

4. **Determine the bearing of C from B:**
   - Calculate the angle θ using the arctangent function:
     \[
     \theta = \arctan\left(\frac{C_{By}}{C_{Bx}}\right)
     \]

   - Convert this angle into a bearing. If the calculated angle is in the standard position, adjust it to a bearing.

Let's compute the exact values using these formulas:

### Calculations:
1. **Coordinates of Hospital B:**
   - \( B_x = 9 \cos(143°) \approx -7.25 \)
   - \( B_y = 9 \sin(143°) \approx 4.96 \)

2. **Coordinates of Hospital C:**
   - \( C_x = 15 \cos(130°) \approx -9.64 \)
   - \( C_y = 15 \sin(130°) \approx 11.51 \)

3. **Relative position of Hospital C from Hospital B:**
   - \( C_{Bx} = C_x - B_x \approx -9.64 - (-7.25) \approx -2.39 \)
   - \( C_{By} = C_y - B_y \approx 11.51 - 4.96 \approx 6.55 \)

4. **Calculate the bearing of C from B:**
   - \( \theta = \arctan\left(\frac{C_{By}}{C_{Bx}}\right) = \arctan\left(\frac{6.55}{-2.39}\right) \approx -69.7° \)

Since this angle is negative and we want the bearing:
   - Add 180° to the angle because it lies in the second quadrant.
   - Bearing = 180° - 69.7° = 110.3°

Thus, the bearing of Hospital C from Hospital B is approximately 110.3°.

Would you like any more details or have any questions?

### Relative Questions:
1. How do you convert a bearing to an angle in the standard position?
2. Why do we use trigonometric functions to find coordinates in bearing problems?
3. What is the difference between a bearing and an angle in standard position?
4. How do you calculate the relative position between two points?
5. Why do you need to adjust the angle obtained from arctan to find a bearing?
6. What quadrants affect the calculation of bearings?
7. How would you determine distances between hospitals using trigonometry?
8. What role does the unit circle play in bearing calculations?

### Tip:
When dealing with bearings, always ensure to convert angles correctly and adjust your final angle based on the quadrant it falls in to get an accurate bearing.

Learn how to calculate the bearing of Hospital C from Hospital B using trigonometric functions and vector addition. This problem involves applying trigonometry in real-world contexts, specifically in determining angles and directions between points. Suitable for students in Grades 10-12.

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