Worked: Hc and Zn by formula, 3 December 2026
1The lesson in plain words
With latitude, declination and LHA, two formulas give Hc and Zn. Worked: a morning Sun sight in mid-Atlantic on 3 December 2026.
The formulas
sin Hc = sin Lat × sin Dec + cos Lat × cos Dec × cos LHA. Use N positive and S negative for both latitude and declination. For the azimuth: cos Z = (sin Dec − sin Lat × sin Hc) ÷ (cos Lat × cos Hc). This Z runs from 0° to 180°, measured from north. Use degree mode. Take inverse sine for Hc and inverse cosine for Z. Keep full calculator precision until the final rounding.
The sight
On 3 December 2026, your DR is 16°40′N 038°20′W. At 1142:18 UT, your corrected Sun altitude is Ho 35°00.4′. For that UT, the almanac gives GHA 358°08.1′ and Dec S22°08.1′ (Celestial: time and the almanac).
The AP
Use the same AP as the tables need: latitude 17°00′N, longitude 038°08.1′W. LHA = GHA − west longitude = 358°08.1′ − 38°08.1′ = 320°. LHA is more than 180°, so the Sun is east of your meridian, as it should be in the morning.
Hc
sin 17° × sin (−22°08.1′) = −0.1102. cos 17° × cos 22°08.1′ × cos 320° = +0.6786. The sum is 0.5684, so Hc = 34°38.4′. Intercept = 35°00.4′ − 34°38.4′ = 22.0 M towards.
Zn
The cosine formula gives Z = 133.6°. The body is east of you (LHA over 180°), so Zn = Z = 133.6°T. When LHA is under 180°, the body is west of you and Zn = 360° − Z. Check for sense: a December morning Sun at 17°N rises south of east and swings south.
From the DR
With a calculator, you can use the DR directly instead of choosing a whole-degree AP. From the DR: LHA 319°48.1′, Hc 34°43.9′, Zn 133.3°, intercept 16.5 M towards. Plotted from the DR, this gives practically the same position line as the AP did.
2Key data
- Altitude
- sin Hc = sin Lat sin Dec + cos Lat cos Dec cos LHA; S negative
- Azimuth
- cos Z = (sin Dec − sin Lat sin Hc) ÷ (cos Lat cos Hc)
- Naming Zn
- LHA > 180°: Zn = Z; LHA < 180°: Zn = 360° − Z
- Worked inputs
- AP 17°N, LHA 320°, Dec S22°08.1′, Ho 35°00.4′
- Worked result
- Hc 34°38.4′, Zn 133.6°T, intercept 22.0 M towards
3Key points
- 1
sin Hc = sin Lat sin Dec + cos Lat cos Dec cos LHA; give south values a minus sign.
- 2
Z from the cosine formula is from north: body east (LHA > 180°) Zn = Z; west, Zn = 360° − Z.
- 3
Take GHA and Dec for the exact UT: 4 seconds of time is 1′ of longitude.
- 4
A calculator can reduce from the DR: different intercept, practically the same line.
4Frequent mistakes
Wrong: LHA = GHA + west longitude.
Right: Subtract west longitude, add east: LHA = GHA − W or + E, then keep it between 0° and 360°.
Wrong: Enter S22°08.1′ as a positive number when your latitude is north.
Right: Enter north latitude and declination as positive, south as negative. Apply each sign independently. Here Dec is −22.135°; using +22.135° gives Hc near 52°.
Wrong: The Z from the formula is the true azimuth.
Right: Z runs 0°–180° from north. Name it from LHA: body west (LHA under 180°) means Zn = 360° − Z.
5Practice case & exam keys
Same sight. A second crew member reduces it from the DR, 16°40′N 038°20′W, and gets 16.5 M towards. The navigator, using AP 17°00′N 038°08.1′W, got 22.0 M towards. One of them must be wrong, says the skipper. Is that right?
Show the answer
No. The intercept depends on where you start. From the DR: LHA 319°48.1′, Hc 34°43.9′, Zn 133.3°, 16.5 M towards. Each intercept, laid off from its own starting point, ends on practically the same position line. Compare the plotted lines, not the intercepts.
Exam keys
- sin Hc = sin L sin d + cos L cos d cos LHA; use N positive and S negative for both L and d.
- cos Z = (sin d − sin L sin Hc) ÷ (cos L cos Hc); LHA > 180° → Zn = Z, else 360° − Z.
- LHA = GHA − west longitude, + east longitude.
- Worked 3 Dec 2026: Hc 34°38.4′, Zn 133.6°T, intercept 22.0 M towards.
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