...consequently A' + B' < 180° (Art. 13G) ; that is, 360°— (a + 5) < 180°, or a + b > 180°. 138. The sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. Thus let ABC be any spherical triangle, of which the angles are A, B, and C, and the sides opposite...

...sin b sin A = sin a sin Д or sin 6 : sin a :: sin В : sin A, that is, — The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. Other relations may be found by making other combinations of the vector axes in equation (3).*...

...circles have a pole and its polar in common, prove that they have a common radical axis. 3. Prove that the sines of the sides of a spherical triangle are proportional to the sines of its opposite angles. 4. Find on a sphere the locus of a point from which the tangents to two lesser...

...39 32. Geometrical Proof of the Rule of Sines.— To show geometrically that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. Let ABC (fig. 16) be a spherical triangle, 0 the centre of the sphere. From any point P in 0C...

...of an isosceles spherical triangle are equal. 54. Show geometrically that the sines of the angles, of a spherical triangle are proportional to the sines of the opposite sides. Given A = 136°, a = 155° 55', 6 = 144° 45', find B. (35.) 55. In a spherical triangle ABC,...

...abacus. P. 170: »GEBER IBN APHLA seems to have discovered the theorem that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides». M. SUTER (Zur Geschichte der Trigonometrie; Biblioth. Mathem. 1893, p. 7) a fait observer...

...trigonometry. From them are or can be derived all other general equations relating to spherical triangles. 226. The sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. The equations expressing this theorem are thus deduced from [152] : From the first and second of those...

...triangle, prove sin* I = sin' ? + sin*-. (45) Case II. — Two Sides and the Angles opposite to them. 29. The sines of the sides of a spherical triangle are proportional to the sines of their opposite angles. DEM. — From equations (16), (19) we get, by multiplication, 2 sin %A cos£...

...also be written in the form of proportions sin a : sin b : sin с = sin A : sin B : sin C. That is, the sines of the sides of a spherical triangle are proportional to the sines of the opposite angles. be employed in the above proof instead of sin A, sin B, or sin C. These sines, however, are equal to...

...[II, theor. 8. 1. If the theorem be regarded as relating to a spherical triangle, it may be written : The sines of the sides of a spherical triangle are proportional to the sines of the opposite angles ; and the law of cosines may be expressed in like form. QUESTIONS. If ABC be any spherical triangle,...