13. cos C =- cos A cos B + sin A sin B cos c. 14. cot a sin b = cot A sin C + cos C cos b. 15. cot a sin c = cot A sin B+ cos B cos c. 16. cot b sin a cot B sin C+ cos C cos a. = 17. cot b sin c = cot B sin A + cos A cos c. 18. cot c sin a 21. sin a cos C sin b cos c = = sin b cos c cos A (Art. 194) cos b sin c cos A. sin a cos c cos B. sin (s — b) sin (s sin b sin c sin s sin (sa). sin b sin c sin (s - b) sin (s—c). 2√sin s sin (s — a) sin (s — b) sin (s – c) 2n sin b sin c where n = √sin s sin (s — a) sin (s — b) sin (s — c). 32. tan a = 33. sin a 2√-cos S cos (S-A) cos (S-B) cos(S-C) sin B sin C [sin(s-a)+sin(s-b)+sin(s-c)-sin s] 2n 46. tan E = √tan stan (s—a) tan (s—b) tan † (s—c). EXAMPLES. 1. Find the time of sunrise at a place whose latitude is Ans. 5h 9m 139.- 2. Find the time of sunset at Cincinnati, lat. 39° 6' N., 3. Find the time of sunrise at lat. 40° 43' 48" N., in the 4. Find the time of sunrise at Boston, lat. 42° 21' N., 5. Find the length of the longest day at lat. 42° 16′ 48′′.3 Ans. 15h 5m 50o. 6. Find the length of the shortest day at New Bruns- 7. Find the hour angle and azimuth of Antares, declina- 8. Find the hour angle and azimuth of the Nebula of 9. Find the azimuth and altitude of Regulus, declination = : 44° 10' 33". 10. Find the azimuth and altitude of Fomalhaut, dec- Ans. Azimuth = S. 27° 18′ 40′′ E.; Altitude = 11° 41' 37". in lat. 11. Find the azimuth and altitude of a star to an observer 12. Find the hour angle (t) and declination (8) of a star 13. Find the distance between Regulus and Antares, the Ans. 99° 55' 44".9. 14. Find the distance between the sun and moon when Ans. 89° 52′ 55′′.5. 15. Find the shortest distance on the earth's surface, in Ans. 2562 miles. 16. Find the shortest distance on the earth's surface Ans. 6444 nautical miles. 17. Given the right ascension of a star 10h 1m 98.34, and Ans. Latitude = ; Longitude = 18. Given the obliquity of the ecliptic w, and the sun's = 19. Given the obliquity of the ecliptic 23° 27' 18".5, and the sun's longitude 59° 40' 1".6; to find his right ascension (a), and declination (8). Ans. a 3h 49m 52.62; &= 20° 5′ 33′′.9 N. 20. Given the sun's declination 16° 0′ 56′′.4 N., and the obliquity of the ecliptic 23° 27' 18".2; to find his right ascension (a), and longitude (λ). Ans. a 9h 14m 193.2; λ = = 136° 7' 6".5. 21. Given the sun's right ascension 14h 8m 19.06, and the obliquity of the ecliptic 23° 27' 17".8; to find his longitude (A), and declination (8). Ans. λ= 214° 20' 34".7; 8 12° 58' 34".4 S. = 22. Given the sun's longitude 280° 23′ 52".3, and his declination 23° 2' 52".2 S.; to find his right ascension (a). Ans. a 18h 45m 148.7. 23. In latitude 45° N., prove that the shadow at noon of a vertical object is three times as long when the sun's declination is 15° S. as when it is 15° N. 24. Given the azimuth of the sun at setting, and also at 6 o'clock; find the sun's declination, and the latitude. 25. If the sun's declination be 15° N., and length of day four hours, prove tan & sin 60° tan 75°. = 26. Given the sun's declination and the latitude; show how to find the time. when he is due east. 27. If the sun rise northeast in latitude 4, prove that cot hour angle at sunrise - sin . 28. Given the latitudes and longitudes of two places; find the sun's declination when he is on the horizon of both at the same instant. 29. Given the sun's declination 8, his altitude h at 6 o'clock, and his altitude h' when due east; prove sin2 8 sin h sin h'. = |