...not found in this line of study one example involving the relation of cause and effect. We may say of any triangle, "The sides are proportional to the sines of the opposite angles," but we cannot properly say any part of a triangle is the cause of any other part. Whoever can speak...

...tany tana + tanatan/3 = l. CHAPTER XIII. RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE. 137. In any triangle the sides are proportional to the sines of the opposite angles; that is, ab _ c sin A sin B sin G' (1) Let the triangle AEG be acute-angled. From A draw AD perpendicular...

...When complement of В is middle. § 3. PRINCIPLES. THEOREM I. In any spherical triangle, the sines of the sides are proportional to the sines of the opposite angles. Let ABC be a spherical triangle on a sphere whose centre is 0. From any point in О В let fall a perpendicular...

...the triangle of forces XYZ, the angle the angle and the angle Z = C + D. XlC, And further, because in any triangle the sides are proportional to the sines of the opposite angles, then H Fs1 ~Y sin Z sin X and putting in the equivalents of the angles, we have HT? 1 1? 1 ri • r<>...

...the angles opposite these sides by A, B, C, respectively. a. Theorem of Sines. § 65. In any plane triangle the sides are proportional to the sines of the opposite angles. Let ABC represent any plane triangle. Let p represent a perpendicular let fall from any vertex upon the opposite...

...formulae from a direct consideration of the solid angle itself. 3. In a spherical triangle, the sines of the sides are proportional to the sines of the opposite angles. Let ABC, Fig. 1, be a spherical triangle, 0 the center of the sphere. The angles of the triangle are the inclinations...

...Similarly it may be proved that, b = c cos A + a cos С ; c = а cos В + b cos A 106. III. 2"o proce that, in any triangle, the sides are proportional to the sines of the angles opposite; or, To proce that abc sin A sin В sin С From .4, any one of the angular points,...

...expressions for all the values of a, cosai (1 — Cos Í) = sin2!» (1 + cos <). 0) (2) 5. Prove that in any triangle the sides are proportional to the sines of the opposite angles. If in a triangle ABC perpendiculars are drawn from the vertices to the opposite sides, show that the...

...or b : с — sm В : sIn С ; с sin С с - sin С а ~ sin A1 or с : а = sin С : sin Rule. — In any triangle, the sides are proportional to the sines of the opposite angles. Art. 615. RULES USED IN LOGARITHMS. RULES FOH THE CHARACTERISTIC. I. For a number greater than 1 the...

...(J. + B) a + b 52 i ¿i _ az III. Law of Cosines, cos A = — ?"— - , etc. ¿be 59. Law of Sines. In any triangle the sides are proportional to the sines of the angles opposite. Let ABO be any triangle, p the perpendicular from В on b. In I (Fig. 34), 0 is an...