. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . parts of a spherical triangle may be com-puted from the above data. EXAMPLES. 1. In the spherical triangle whose angles are A, B, C,prove B + C-A<tt (1) C+A-B<tt (2) A + B - C < 7T . (3) 2. If C is a right angle, prove A + B < f 7T (1), and A - B < ^ (2). It 3. The angles of a triangle are A, 45°, and 120°; find themaximum value of A. Ans. A < 105°. 4. The angles of a triangle are A, 30°, and 150°; find themaximum value of A. Ans. A < 60°. 5. The angles
. A treatise on plane and spherical trigonometry, and its applications to astronomy and geodesy, with numerous examples . parts of a spherical triangle may be com-puted from the above data. EXAMPLES. 1. In the spherical triangle whose angles are A, B, C,prove B + C-A<tt (1) C+A-B<tt (2) A + B - C < 7T . (3) 2. If C is a right angle, prove A + B < f 7T (1), and A - B < ^ (2). It 3. The angles of a triangle are A, 45°, and 120°; find themaximum value of A. Ans. A < 105°. 4. The angles of a triangle are A, 30°, and 150°; find themaximum value of A. Ans. A < 60°. 5. The angles of a triangle are A, 20°, and 110°; find themaximum value of A. Ans. A < 90°. 6. Any side of a triangle is greater than the differencebetween the other two. RIGHT SPHERICAL TRIANGLES. 185. Formulae for Right Triangles. — Let ABC be a spherical triangle in which C isa right angle, and let 0 be thecentre of the sphere; then willOA, OB, OG be radii: let a, b, cdenote the sides of the triangle O*opposite the angles A, B, C, re-spectively ; then a, b, and c arethe measures of the angles BOC,COA, and RIGHT SPHERICAL TRIANGLES. 271 From any point D in OA draw DE _L to OC, and from Edraw EF _L to OB, and join DF. Then DE is _L to EF ( 537). Hence (Geom. Art. 507), DF is 1_ to OB ; .-. Z DFE = Z B . (Art. 183) AT OF OF OE , , • . ~ N Now = : that is, cos c = cos a cos b . (1) OD OE OD v ; DE = DEJDF h . , sin5 = sinBsinc . (2)OD DF OD v Interchanging as and 6s, sin a = sin A sin c . (3) — = — . —: that is, tan a= cos B tan c . (4)OF DF OF J v Interchanging as and Us, tan b = cos A tan c . (5) = ; that is, tan b = tan B sin a . (6) OE EF OE w Interchanging as and &s, tan a = tan A sin b . (7)Multiply (6) and (7) together, and we get tanAtanB = — = -1-, by (1) cos a cos b cos c .?. cos c = cot A cot B (8) Multiply crosswise (3) and (4), and we getsin a cos B tan c = tan a sin A sin c. .-. cosB = = sin A cos b, by (1) ... (9) cos a J v J v J Intercha
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Keywords: ., bookcentury1900, bookdecade1900, booksubjecttrigono, bookyear1902
