point. Azimuth is measured in degrees from 0° to 360°.The conventional symbol for azimuth is the letter A orZ.Astronomical TriangleThe solutions of problems involving the threecoordinate systems are made by means of sphericaltrigonometry. A figure of prime importance is thespherical triangle that lies on the celestial sphere andwhose vertices are the pole, the zenith, and thecelestial body involved. This is known as theastronomical or the PZS (pole-zenith-star) triangle.The astronomical triangle is shown in figure 15-3.As in the case of all spherical triangles, the sides canbe expressed as the angles subtended at the center ofthe sphere. In the astronomical triangle, the sidebetween the pole and the zenith is the colatitude(90° - 0), between the pole and the star is thecodeclination or polar distance (90° - 6),and betweenthe zenith and the star is the coaltitude, or zenithdistance (90° - h). The angle at the zenith is theazimuth angle (A) of the body. The angle at the poleis the hour angle (t). The angle at the star is known asthe parallactic angle and is little used in computa-tions. If the three elements of the astronomic triangleare known, the others can be found by means of spheri-cal trigonometry. The fundamental equation is the lawof cosines. cos a = cos b cos c + sin b sin c cos a, inwhich a, b, and c are the sides of a spherical triangle,and A is the angle opposite side a (B and C are theangles opposite sides b and c, respectively). Allformulas required for the solution of the astronomictriangle may be derived from this law of cosines.Astronomical TablesUsed by SurveyorsThe declination and Greenwich hour angle of thesun, moon, and selected planets are given for every evenhour of GMT for everyday in the year in the daily pagesof the Nautical Almanac and the Ephemeris of the Sun,Polaris, and Other Selected Stars. These publicationsare prepared by the U.S. Naval Observatory and areavailable for sale at the U.S. Government PrintingOffice, Washington, D.C. Condensed tables of data arealso available from various manufacturers of surveyingequipment.Suppose that you want to determine the GHA anddeclination of the sun for an observation made at zonetime l&23m18’ on 17 May 1986 in longitude79”37’12”W.The ZD is +5; therefore, GMT Of theobservation was 15~23m18’.Table 15-1 shows the relevant daily page of the 1986Nautical Almanac. You can see that for 15~~O& on 17May the GHA listed for the sun is 45054.8’. For the extra23”’18S you turn to a table of increments andcorrections in the back of the book. Table 15-2showsthe relevant page of the table. Under 23m and beside 18^{s}in the Sun column you find an increment of 5049.5’. TheGHA of the sun at the time of observation, then, was45°54.8’ + 5049.5’,or 51044.3’.On the daily page of 17 May (table 15-1), theNautical Almanac gives a sun declination for 15%0’’’00’GMT on N 19°21.3’. At the foot of the column, yousee a small d and the figure 0.6. In the increments andcorrections table (table 15-2), you see a column of vor d corrections for declination. You go down thiscolumn to the figure 0.6, where you find that the dcorrection in this case is 0.2'. Whether you add thiscorrection or subtract it depends upon whether thedeclination of the sun is increasing or decreasing withtime. A glance at the daily page shows that in this case,it is increasing; therefore, the declination of the sun atthe time of observation was N 19°21.3’ + 0.2’, orN 19021.5’.On an opposing daily page of the Nautical Almanac(table 5-3), the declinations of a select list of 57prominent stars are given. Instead of the GHAs of thesestars, however, the sidereal hour angle (SHA) of eachstar is given. The sidereal angle of a star is its arcdistance westward from the vernal equinox or first pointof Aries. The GHA of a star is its arc distance westwardfrom the hour circle of the first point of Aries.For GHA of a star, you first determine GHA of thefirst point of Aries in the same manner described for thesun. You can see Aries listed in tables 15-2 and 15-3.You then add this to the SHA of the star, as given in thedaily page of the Nautical Almanac (table 15-3). Thiscan be stated as follows: GHA star= GHA Aries + SHAstar. If the result is greater than 360°, you subtract 360°from it.For declination of a star, you use the declinationlisted on the daily page; this is good for a star at anytimeof the day.NOTE: The SHA and GHA of the vernal equinoxare factors used in star observations; however, neither isapplicable in observation of navigational plaints.15-7