...+ CB : AC— CB:: tangí (B+C) : tang-i (B—C) it follows that in any triangle the sum of any two sides is to their difference, as the tangent of half the sum of the two angles opposite these sides, is to the tangent of half the difference of these same angles. Let...

...angle C. LIX. In any triangle (supposing any side to be the basr, and calling the other two the sides) the sum of the sides is to their difference, as the...tangent of half the sum of the angles at the base is to the tangent of half the difference of the tame angles. Thus, in the triangle ABC, if we call...

...chords of double their opposite angles. PROPOSITION IV. (E) 1. In any plane triangle, the sum of any two sides is to their difference, as the tangent of half the sum of their ^opposite angles, is to the tangent of half their difference. Let ABC be any triangle; make BE...

...triangle (supposing any aide to be the base, and calling the other two the tide*) the sum of the sida is to their difference, as the tangent of half the sum of tht ongfcs at the base is to the tangent of half the difference of the tame angla. Thus, in the triangle...

...^r;» ^'otn which tang i (a' -f- 6') sin a' + sin 6' we infer, that the sum of the sines of two arcs is to their difference, as the tangent of half the sum of these arcs is to the tangent of half their difference, is obtained immediately by a very elegant geometrical...

...tangent of half their difference. Let ABC be a plane triangle, the sum of any two sides AB, AC will be to their difference as the tangent of half the sum of the angles at the base ABC, ACB, to the tangent of half their difference. About A as a centre, with AB the greater side for...

...plane triangle are as the sines of the opposite angles. (73.) The sum of two sides of a plane triangle is to their difference as the tangent of half the sum of the opposite angles to the tangent of half their difference. •* ^74.) Formulae for the sine, cosine,...

...first of these cases is shewn to depend on the theorem, that, " the sum of two sidi\s of a triangle is to their difference, as the tangent of half the sum of the opposite angles to the tangent of half their difference." This half difference added to half the sum,...

...should obtain, THEOREM. 44. In any plane triangle, the sum of tfte two sides containing either angle, is to their difference, as the tangent of half the sum of the other two angles, to the tangent of half their difference. Let ABC (PI. I. Fig. 3) be a triangle ;...

...D, E, F, then will ^ ,P_F_ AD AE AF 10. When the vertical angle of any plane triangle is bisected ; the sum of the sides is to their difference as the tangent of the angle made by the bisecting line and the base is to the tangent of half the vertical angle. 11....