...— = -r— : therefore, f sin a sin 4 sin A sin 11 sine , . sin a sin b sin c ' ' ' ' * "' Hence, the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. 21 . Draw CE and DF, respectively perpendicular and parallel to OB; then will the angle...

...let the radius of the sphere be conceived to be continually increased. The consequence will be, that the angles subtended at the centre of the sphere by the sides will be continually diminished, and therefore the sides expressed in degrees will be also continually...

...polar triangle throughout the following treatise. MATHEMATICAL SCIENCES— No. XIII. (1.) To show that the Sines of the Angles of a Spherical Triangle are proportional to the Sines of the opposite sides. Let ABC be the triangle, 0 the centre of the sphere, join OA, OB, 00 ; through A draw...

...opposite sides. We will give an independent proof of this proposition in the following article. 42. The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. Let ABC be a spherical triangle, O the centre of the sphere. Take any point P in OA,...

...given radius. PURE MATHEMATICS REV. CANON HEAVISIDE, MA Third Paper. — 11 1 AM to 1| PM 1. Show that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. In any spherical triangle, a, b, c being the sides subtending the angles A, B, C respectively,...

...expression, namely, »/(! — cos2 a — cos2 6 — cos2c + 2 cos a cos b cos c) sin a sin 6 sin c Thus the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. We will give an independent proof of this proposition in the following article. 42....

...X. To find the vertical component. PH = PFsiaFPG = 2co» sin SP V sin FPG = 2 CD v sin VS sin F. But the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides : .'. sin VS sin V = sin AS sin A = cos X sin NP V = cos X sin 6: .'. PH = 2 n> v cos...

...proportion to four right angles that tlie area of Hie triangle bears to the surface of tlie hemisphere. The sines of the angles of a spherical triangle are...the sides to which they are respectively opposite. PROBLEM FIRST. — To compute approximately the angles subtended by arcs on the earth's surface, and...

...have the following result, sin. g sin. A sin. B sin. C (I5) sin. a sin. b sin. c 331. This proves that the sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. 332. Again, add 1 to both sides of the first equation in (14), and remember that 1...

...spherical triangle is greater than half the spherical excess. OPPOSITE SIDES AND ANGLES. 882. Theorem — The sines of the angles of a spherical triangle are proportional to the sines of the opposite sides. Let ABC be the spherical triangle, and 0 the center of the sphere. From any point P...