To determine the bearing of town P from town Q given that the bearing of town Q from town P is [tex]\(345^\circ\)[/tex], follow these steps:
1. Understanding Bearings:
- Bearings are typically measured clockwise from the north direction.
- Given the bearing of town Q from town P as [tex]\(345^\circ\)[/tex], this means if you stand at town P, you have to turn [tex]\(345^\circ\)[/tex] clockwise from the north to face town Q.
2. Reverse Bearing Calculation:
- To find the bearing of town P from town Q, you essentially need to find the direction you would face if you were at town Q looking back towards town P.
- This involves adding or subtracting [tex]\(180^\circ\)[/tex] because a straight angle (the angle directly opposite) is [tex]\(180^\circ\)[/tex].
3. Adjusting for Bearing Range (0 to 360 degrees):
- Bearings must be within the range from [tex]\(0^\circ\)[/tex] to [tex]\(360^\circ\)[/tex].
- If the resulting bearing exceeds [tex]\(360^\circ\)[/tex], you subtract [tex]\(360^\circ\)[/tex].
- If the resulting bearing is less than [tex]\(0^\circ\)[/tex], you add [tex]\(360^\circ\)[/tex].
4. Performing the Calculation:
- First, add [tex]\(180^\circ\)[/tex] to [tex]\(345^\circ\)[/tex]:
[tex]\[
345^\circ + 180^\circ = 525^\circ
\][/tex]
- Since [tex]\(525^\circ\)[/tex] exceeds [tex]\(360^\circ\)[/tex], subtract [tex]\(360^\circ\)[/tex] to bring it within the standard bearing range:
[tex]\[
525^\circ - 360^\circ = 165^\circ
\][/tex]
So, the bearing of town P from town Q is [tex]\(165^\circ\)[/tex].
