Hence, log a log a+log sin 0-10; the subsidiary angle being previously determined from the equation, a cos +b sin ê, is not adapted to logarithmic computation. Let it be transformed into x=b (= cos 0+sin ) : Hence, log tan being found from the equation, log a +10—log b, we shall have, log a log blog sin (+)-log cos p. Other instances of the use of subsidiary angles will occur in the solutions of plane and spherical triangles. SECTION IV. PLANE TRIGONOMETRY. (46.) A plane triangle consists of six parts, viz :three sides and three angles: if any three of these parts except the three angles be given, the other three become known by the application of principles, which constitute Plane Trigonometry. (47.) Let A B C (fig. 7.) be a triangle, right angled at C; and let the sides opposite to the angles A, B, C, be denoted by the small letters a, b, c, respectively. From the point A as a centre, with radius=1, describe the circle G H, and draw G K perpendicular to AC. By the similar triangles A G K, A BC, AG: GK:: AB: BC, or, 1 sin A::c: a, : A+B=2, and c2=a2+b2, are sufficient to solve every case of right angled plane triangles. We shall present them in one view, and apply them to a few examples. Example I. Suppose the base of a right angled triangle to be 540 yards, and the adjacent acute angle 56° 40′; required the hypothenuse, the perpendicular, and the other acute angle. Here A and b are given, to find a, B, and e. ... log clog b+10-log cos A. log 540.........= 2. 7323938 Referring to the table of logarithms, c=982.6955 yards. Also from equ. (3.), a=b tan A, ... log a log. b+log. tan A-10. By reference to the table, a 821 03 yards. Example II. Given the hypothenuse 58. 45, and the base 31. 5, to find the angles and the perpendicular. Here c 58.45, and b=31 . 5. log cos A........= 9· 7315261 referring to the logarithmic table of sincs, we find A=57° 23′ 23′′. log sin 57° 23′ 23′′ = 9.9254955 The side a however, may be found independently of the angle A, by equ. (5), c2=a2+b2, or, a2=c2—– b2; and, as the numbers b and c consist of several figures, the calculation may be thus facilitated: since c3—b2=(c+b). (c—b), we have 2) 3·3845600 log a or 49. 2364... = 1 · 6922800, as before. Example III. Given a 759.4 and b=33. 29 to find c. |