Part III · The line of position · Page 8 of 8
Sight reduction by calculator
The tables are the navigational triangle solved in advance. A scientific calculator can solve it on the spot, with two formulas. Many professional navigators work this way, and some find it easier than the tables: there is no interpolation, and no assumed position.
sin Hc = sin L · sin d + cos L · cos d · cos LHA
cos Z = (sin d − sin L · sin Hc) ÷ (cos L · cos Hc)
- Work from your DR. L is your DR latitude and LHA uses your DR longitude, minutes and all. The intercept is laid off from the DR.
- South is negative. Put a minus sign on a south latitude or a south declination.
- Z is from north, 0° to 180°. LHA over 180° (the body is east of you): Zn = Z. LHA under 180° (west): Zn = 360° − Z. The same rule in both hemispheres.
- Degrees, not radians. Check that sin 30 gives 0.5 before you start.
The same sight as on the last page, by formula:
DR N 41°12′, W 064°30′. GHA 36°49.8', Dec N 18°40.0', Ho 57°24.4'.
LHA = 36°49.8' − 64°30.0′ + 360° = 332°19.8' (no rounding).
sin Hc = 0.65869 × 0.32006 + 0.75241 × 0.94740 × 0.88564 = 0.21082 + 0.63131 = 0.84213. Hc = sin⁻¹ 0.84213 = 57°22.0'.
cos Z = (0.32006 − 0.55471) ÷ 0.40575 = -0.57829. Z = 125.3°.
LHA over 180°: Zn = Z = 125°.
Intercept: Ho 57°24.4' − Hc 57°22.0' = 2.4 nm toward, from the DR.
A different intercept from a different starting point, but the same line: plotted, the two lines are 0.4 nm apart. In the practice for this module and for planets and the Moon, choose Formula (calculator) above a sight to work it this way. The navigator’s calculator beside every sight also has an Hc and Z tab that does the whole thing, for checking your work.