...angle ; that is, double the sine of that angle measured in the circle ; therefore the sides of the triangle are to each other as the. sines of the opposite angles measured in the same circle, and consequently as the sines of the same angles measured in a circle...

...c=log a + 10— cos 6: from this c=760.129. (48.) The sides of a plane triangle are proportional to the sines of the opposite angles. Let ABC be a plane triangle (fig. 8.), and BD the perpendicular from B upon the side AC ; then by (470, BD=AB . sin A, also BD=BC...

...secant of one of them takes, may be given to the cosine, cotangent, or cosecant of the other. 2 Th. The sides of a plane triangle are to each other as the sines of the angles opposite to them. In right angled triangles this proposition is obvious: for if the subtense...

...angle ; that is, double the sine of that angle measured in the circle ; therefore the sides of the triangle are to each other as the sines of the opposite angles measured in the same circle, and consequently as the sines of the same angles measured in the circle...

...rfe-f-the angle i«c = 90°; as also bed-^-lhe side ac = 90°. The sines of the sides of a spherical triangle are to each other as the sines of the opposite angles. Let abc (fig. 1 14) be a spherical triangle, whose sphere has its centre in o, and unity for radius. If now...

...-f- the angle bac = 90° ; as also bed -j- the side ac = 90°. The sines of the sides of a spherical triangle are to each other as the sines of the opposite angles. Let abc (Jig. 1 14) be a spherical triangle, whose sphere has its centre in o, and unity for radius. If now...

...lines over rugged and uneven ground? 11. The fundamental proposition in plane trigonometry is that the sides of a plane triangle are to each other as the sines of their respective opposite angles; prove this, and state its application to land surveying. 12. Give...

...problem. The rules for both problems are expressed by the following THEOREM : — The Sides of any Plane Triangle are to each other as the Sines of the opposite Angles. NOTE. Since, by the rule, we find the sine of the required angle, and not the angle itself, and since...

...BD : : sine D : sine A ; and therefore, also, BA AD : : sine D : sine B. The above proposition, that the sides of a plane triangle are to each other as the sines- of the opposite angles is a fundamental one in trigonometry, and on it are based all the former observations that have been...

...de-|-the angle foze — 90 0 ; as also bed-\-i\\z side ac = 90°. The sines of the sides of a spherical triangle are to each other as the sines of the opposite angles. Let abc (fig. 114) be a spherical triangle, whose sphere has its centre in o, and unity for radius. If now...