...radius. (Art. 11 9.) THEOREM II. ..* 144. In a plane triangle, Jl 3 the sum of any two of the sides, To their difference; So is the tangent of half the sum of the opposite angles, To the tangent of half their difference. Thus the sum of AB and AC (Fig. 25.) is to...

...180°, the remainder will be the sum of th£ other two angles. Then say, — As the sum of the given sides, Is to their difference ; So is the tangent of half the sum of the remaining angles, To the tangent of half their difference. Half the difference added to half the sum...

...AC; whence the proposition is manifest. PROP. XI. 1 1 . As the sum of the sines of two unequal arcs, is to their difference, so is the tangent of half the sum of those two arcs, to the tangent of half their difference. Let AE and AB be two unequal arcs, of which...

...wholes are as their halves, ie AH : IH : : CE : ED, that is, as the snm of the two sides AB and BC, is to their difference ; so is the tangent of half the sum of the two unknown angles A and C, to the tangent of half their difference. QED THEOREM HI. In any right-lined...

...same angle*. Thus, in the triangle ABC, if we call AB the base, it will he as the sum of AC and CB is to their difference, so is the tangent of half the sum of the angles ABC, BAG', to the tangent of half their difference. Dem. With the longest leg CB as radius,...

...of either angle to the co-tangent of the other angle. As the sum of the sines of two unequal arches is to their difference, so is the tangent of half the sum of those arches to the tangent of half their difference : and as the sum of their co-sines a to their...

...of either angle to the co-tangent of the other angle. As the sum of the sines of two unequal arches is to their difference, so is the tangent of half the sum of those arches to the tangent of half their difference : and as the sum of their co-sines is to their...

...triangle. Or the angles opposite the given sides may be determined as follows. As the sum of the given sides is to their difference, so is the tangent of half the sum of their opposite angles to the tangent of half the difference of the same angles, (Trig. Prop. 6.) And...

...solution of this CASE depends on the following PROPOSITION. In every Plane Triangle, as the sum of any two Sides is to their difference, so is the Tangent of half the sum of the two opposite Angles to the Tangent of half the difference between them. Add this half difference to...

...the tame angla. Thus, in the triangle ABC, if we call AB the base, it will be as the sum of AC and CB is to their difference, so is the tangent of half the sum of the angles ABC, ВАС, to the tangent of half their difference. Dan. With the longest leg CB as radius,...