True and Compass Bearings

A bearing tells you a direction as an angle. A true bearing always starts facing north and turns clockwise, the way a clock's hands move, until you face the point. From A, you turn sixty five degrees to face B, so the bearing of B from A is zero six five degrees. To face C you turn much further, past east and south, all the way round to two hundred and forty degrees. True bearings are always written with three figures, so five degrees is written zero zero five.

A compass bearing starts at north or south, whichever is closer, and turns towards east or west. N forty degrees E means face north, then turn forty degrees towards east. That is the same as the true bearing zero four zero. S thirty degrees W means face south, then turn thirty degrees towards west. South is one hundred and eighty, so this is one hundred and eighty plus thirty, which is two hundred and ten degrees. In the same way, N seventy degrees W is three hundred and sixty minus seventy, which is two hundred and ninety.

If B is on a bearing of zero six five degrees from A, what is the bearing of A from B? Draw a north line at B as well. The two north lines are parallel, so the sixty five degree angle at A appears again at B, measured from south: these are alternate angles. Facing A from B, you turn from north through south, which is one hundred and eighty degrees, and then sixty five more. So the back bearing is two hundred and forty five degrees. The rule: add one hundred and eighty if the bearing is under one hundred and eighty, otherwise subtract it.

A walker goes twelve kilometres east and then five kilometres north. East and north are at right angles, so the start, the turning point and the finish make a right-angled triangle, and the straight-line distance back to the start is its hypotenuse. By Pythagoras, d squared equals twelve squared plus five squared, which is one hundred and forty four plus twenty five, or one hundred and sixty nine. So d is thirteen kilometres. Any two legs on bearings that differ by ninety degrees make a right angle in the same way.

Now find the bearing too. A ship sails four kilometres north, then four root three kilometres east. The distance from the start is root of sixteen plus forty eight, which is root sixty four, or eight kilometres. For the bearing, you need the angle at the start between north and the line to the finish. From that angle, four root three is opposite and four is adjacent, so tan theta is root three, and theta is sixty degrees. It is measured from north, clockwise, so the bearing of the finish from the start is zero six zero degrees. Quiz and exercises now.