7.1 Bearings, Azimuths, and Direction Conversions

Two Ways to State a Direction

Every line on the FS exam has a direction and a distance. Direction is expressed two ways, and you must move between them fluently. An azimuth is a horizontal angle measured clockwise from a reference meridian, conventionally north, and runs from 0 deg up to (but not including) 360 deg. A quadrant bearing is the acute angle (0 deg to 90 deg) between the line and the nearest meridian, written with the meridian first and the side second: N 30 deg 15 min E, S 62 deg 00 min W, and so on.

The two systems describe the same physical line. A line at azimuth 120 deg points into the southeast quadrant, so its bearing is S 60 deg E (180 deg minus 120 deg). Knowing the quadrant tells you the sign of the coordinate components later, so never skip identifying it.

Bearing-to-Azimuth Conversion

Memorize the four quadrant relationships. With azimuths measured from north:

Worked example. Convert N 47 deg 30 min W to an azimuth. This is the NW quadrant, so Az = 360 deg - 47 deg 30 min = 312 deg 30 min. Reverse check: 360 deg - 312 deg 30 min = 47 deg 30 min, and an azimuth past 270 deg lies in the NW quadrant, so the bearing is N 47 deg 30 min W. The arithmetic and the quadrant agree.

Forward and Back Directions

Every line has a forward direction (A to B) and a back direction (B to A) that differ by exactly 180 deg. For azimuths: if the forward azimuth is less than 180 deg, add 180 deg; if it is 180 deg or more, subtract 180 deg. This keeps the result inside 0 deg to 360 deg.

Example. Forward azimuth A to B is 312 deg 30 min. Back azimuth B to A = 312 deg 30 min - 180 deg = 132 deg 30 min. For bearings, simply flip both letters: the back bearing of N 47 deg 30 min W is S 47 deg 30 min E, which checks against azimuth 132 deg 30 min (SE quadrant, 180 deg - 132 deg 30 min = 47 deg 30 min).

Angle-to-Direction and the Wraparound Trap

In traverse work you often compute the azimuth of the next course from the azimuth of the current course plus a measured interior or deflection angle. The general rule for advancing by an interior angle on a traverse run counterclockwise is: next azimuth = back azimuth of previous course + interior angle. After every addition, normalize: if the result is 360 deg or more, subtract 360 deg; if negative, add 360 deg.

• Working in degrees-minutes-seconds, borrow 60, not 100, when subtracting.
• A negative azimuth is never a final answer; add 360 deg.
• Keep magnetic, grid, and true (astronomic) references separate. Applying a magnetic declination corrects a magnetic bearing to a true bearing; a grid convergence relates true to grid north. FS distractors often pair a correct number with the wrong reference.

Common FS Traps

The classic mistakes are choosing the wrong quadrant formula, forgetting to normalize past 360 deg, and reporting a back azimuth that fell out of range. Always sketch the line in the correct quadrant before trusting a number; a 10-second sketch catches sign errors the calculator cannot.

Why Direction Discipline Pays Off

Direction is the single value that propagates through every later computation in this chapter. The same azimuth that you convert here becomes the cosine that produces a latitude in Section 7.2, the term that closes a traverse in Section 7.3, and the orientation of a curve tangent in route work. A 1-minute error in a poorly converted azimuth at the start of a 1,000 ft course produces about 0.3 ft of position error at the far end (1000 x tan(1 min) is roughly 0.29 ft), enough to fail a tight relative-precision standard. That sensitivity is exactly why the FS rewards careful direction handling.

A few habits keep you out of trouble across an entire problem set:

• Always state the reference. Write whether a direction is true, grid, magnetic, or assumed. The FS reference handbook works in azimuths from north, so converting magnetic field bearings to true before computing is a routine first step.
• Carry full DMS precision. Round only at the final answer. Truncating seconds early accumulates into a visible coordinate error after several courses.
• Use the calculator's atan2 thinking. When you later inverse coordinates back to a direction, the signs of the latitude and departure tell you the quadrant; the bare arctangent does not, because it cannot distinguish a NE line from its SW reciprocal.
• Re-sketch after every angle turn. Interior-angle and deflection-angle traverses are where wraparound errors hide; a running sketch is the cheapest insurance available.

Master these conversions cold, because the rest of the chapter assumes you can move between bearing, azimuth, forward, and back directions without hesitation.