By the other equation in formula (T), we can find the angle C; but, for the sake of variety, we will find the angle C by the application of the third equation in formula (U). To show the harmony and practical utility of these two sets of equations, we will find the angle A, from the equation 2. In a spherical triangle ABC, given the angle A, 38° 19′ 18′′, the angle B, 48° 0′ 10′′, and the angle C, 121o 8′ 6′′, to find the sides a, b, c. Apply proposition 6, spherics. We now find the angles to the spherical triangle, whose sides are these supplements. supp. 46 24 45-a of the original triangle. In the same manner we find b=60° 14′ 25′′ c=89° 1′ 14′′ EXAMPLES FOR EXERCISE. 1. In any triangle, ABC, whose sides are a, b, c, given b=118°2′ 14′′, c=120° 18′ 33′′, and the included angle A=27° 22′ 34′′, to find the other parts. Ans. a=23° 57′ 13′′, angle B=91° 26′ 44′′, and C=102° 5′ 54′′. 2. Given A=81° 38′ 17′′, B=70° 9′ 38′′, and C=64° 46′ 32", to find the sides a, b, and c. Ans. a=70° 4′ 18′′, b=63° 21′ 27′′, and c=59° 16′ 23′′. 3. Given the three sides a=93° 27′ 34′′, b=100° 4′ 26′′, and c=96° 14' 50", to find the angles A, B, and C. · Ans. A=94° 39′ 4′′, B=100° 32′ 19′′, and C-96° 58′ 36′′. 4. Given two sides, b=84° 16′, c=81° 12′, and the angle_C=80° 28', to find the other parts. Ans. The result is ambiguous, for we may consider the angle B as acute or obtuse. If the angle B is acute, then A=97° 13′ 45′′, B=83° 11′ 24′′, and a—96° 13′ 33′′. If B is obtuse, then A=21° 16′ 44′′, B=96° 48′ 36′′, and a=21° 19′ 29′′ *The sine complement of 131° 59′ 50′′, is the same as the sine complement of 48° 0' 10". 5. Given one side, c=64° 26', and the angles adjacent, A=49°, and B-52°, to find the other parts. Ans. b 45° 56′ 46", a=43° 29′ 49", and C=98° 28′ 5′′. 6 Given the three sides, a=90°, b=90°, c=90°, to find the angles A, B, and C. Ans. A 90°, B=90°, and C-90°. 7. Given the two sides, a=77° 25' 11", and c=128° 13' 47", and the angle C, to find the other parts. Ans. b 84° 29' 24", A=69° 14', and B=72° 28' 46'. 8. Given the three sides, a, b, c, a=68° 34′ 13′′, b=59° 21′ 18, and c=112° 16' 32", to find the angles A, B, and C. Ans. A 45° 26' 12", B=41° 11' 6", C=134° 54' 27" APPLICATION. Spherical trigononometry becomes a science of incalculable importance in its connection with geography, navigation, and astronomy; for neither of these subjects can be understood without it; and to stimulate the student to a study of the science, we here attempt to give him a glimpse at some of its points of application. Let the lines in the annexed figure represent circles in the heavens above and around us. Let Z be the zenith, or the point just overhead, Hch the horizon, PZH the meridian in the heavens, P the pole of the earth's equator; then Ph is the latitude of the observer, and PZ is the co.latitude. Qcq is a portion of the equator, and the dotted, curved line, mS'S, parallel to the equator, is the parallel of the sun's declination at some particular time; and in this figure the sun's declination is supposed to be north. By the revolution of the earth on its axis, the sun is apparently brought from the horizon, at S, to the meridian, at m: and from thence it is carried down on the same curve, on the other side of the meridian; and this apparent motion of the sun (or any other celestial body) makes angles at the pole P, which are in direct proportion to their times of description. The apparent straight line, Zc, is what is denominated, in astronomy, the prime vertical; that is, the east and west line through the zenith, passing through the east and west points in the horizon. When the latitude of the place is north, and the declination is also north, as is représented in this figure, the sun rises and sets on the horizon to the north of the east and west points, and the distance is measured by the arc cS, on the horizon. This arc can be found by means of the right angled spherical triangle cqS, right angled at q. Sq is the sun's declination, and the angle Scq is equal to the co.latitude of the place; for the angle cPh is the latitude, and the angle Scq is its complement. The side cq, a portion of the equator, measures the angle cPq, the time of the sun's rising or setting before or after six, apparent time. Thus we perceive that this little triangle cSq, is a very important one. When the sun is exactly east or west, it can be determined by the triangle ZPS'; the side PZ is known, being the co.latitude; the angle PZS' is a right angle, and the side PS' is the sun's polar distance. Here, then, is the hypotenuse and side of a right angled spherical triangle given, from which the other parts can be computed. The angle, ZPS' is the time from noon, and the side ZS' is the sun's zenith distance at that time. FORMULA FOR TIME. The most important problem in navigation is that of finding the time from the altitude of the sun, when the sun's declination and the latitude of the observer are given. This problem will be un derstood by the triangle one hour, will give the time from apparent noon, when any particular altitude, as TS, may have been observed. PS is known by the sun's declination at about the time; and PZ is known, if the observer knows his latitude. Having these three sides, we can always find the sought angle at the pole, by the equations already given in formulas (T), or (U); but these formulas require the use of the co.latitude and the co.altitude, and the practical navigator is very averse to taking the trouble of finding the complements of arcs, when he is quite certain that formulas can be made, which comprise but the arcs themselves. The practical man, also, very properly demands the most concise practical results. No matter how much labor is spent in theorizing, provided we arrive at practical brevity; and for the especial accommodation of seamen, the following formula for finding time has been deduced. From the fundamental equation of spherical trigonometry, taken from page 191 we have, Now, in place of cos.ZS, we take sin. ST, which is, in fact, the same thing, and in place of cos.PZ, we take sin.lat., which is also the same. In short, let A= the altitude of the sun, L= the latitude of the observer, and D= the sun's polar distance. |