Calculating the Included Angle CAB from Whole Circle Bearings

This solution explains how to determine the included angle between two lines, AB and AC, given their respective Whole Circle Bearings (WCB).

Understanding Whole Circle Bearings (WCB)

Whole Circle Bearing is the angle measured in a clockwise direction from the North direction (0 degrees) to the line. Bearings range from 0° to 360°.

Given Bearings

The problem provides the following bearings:

• The Whole Circle Bearing (WCB) of line AB is $18^\circ 15'$.
• The Whole Circle Bearing (WCB) of line AC is $335^\circ 45'$.

Calculating the Included Angle CAB

The included angle CAB is the angle formed at point A, between the lines AC and AB. We can find this angle by finding the difference between the bearings of the two lines.

Method 1: Clockwise Angle from AC to AB

To find the angle measured clockwise from line AC to line AB, we subtract the WCB of AC from the WCB of AB. If the WCB of AB is smaller than the WCB of AC, we add $360^\circ$ to the WCB of AB to ensure a positive angle representing the clockwise measurement.

Angle CAB $= (\text{WCB of AB} - \text{WCB of AC}) + 360^\circ$ (if WCB of AB < WCB of AC)

Angle CAB $= (18^\circ 15' + 360^\circ) - 335^\circ 45'$

Angle CAB $= 378^\circ 15' - 335^\circ 45'$

To perform the subtraction:

• Convert $378^\circ 15'$ to $377^\circ (60+15)' = 377^\circ 75'$.
• Subtract the minutes: $75' - 45' = 30'$.
• Subtract the degrees: $377^\circ - 335^\circ = 42^\circ$.

Therefore, the angle measured clockwise from AC to AB is $42^\circ 30'$.

Method 2: Clockwise Angle from AB to AC

To find the angle measured clockwise from line AB to line AC, we subtract the WCB of AB from the WCB of AC.

Angle BAC $= \text{WCB of AC} - \text{WCB of AB}$

Angle BAC $= 335^\circ 45' - 18^\circ 15'$

Subtract the minutes: $45' - 15' = 30'$.

Subtract the degrees: $335^\circ - 18^\circ = 317^\circ$.

So, the angle measured clockwise from AB to AC is $317^\circ 30'$.

Determining the Included Angle

The included angle between two lines is typically the smaller angle. The angle calculated in Method 2 ($317^\circ 30'$) is a reflex angle. The other angle (the included angle) is found by subtracting this value from $360^\circ$.

Included Angle $= 360^\circ - \text{Angle BAC}$

Included Angle $= 360^\circ 00' - 317^\circ 30'$

To perform the subtraction:

• Convert $360^\circ 00'$ to $359^\circ 60'$.
• Subtract the minutes: $60' - 30' = 30'$.
• Subtract the degrees: $359^\circ - 317^\circ = 42^\circ$.

The included angle is $42^\circ 30'$.

Conclusion

Both methods confirm that the included angle CAB is $42^\circ 30'$.