substituting these for their equals in the preceding equation it becomes As the angles have each the same relations to the corresponding sides of a triangle, the same formula by a proper modification will furnish the values of the angles A and c. It may be expressed in ordinary language thus, the sine of half either angle of a triangle is equal to radius into the square root of half the sum of the three sides minus one of the adjacent sides, into half the sum minus the other adjacent side, divided by the rectangle of the adjacent sides. To apply this to an EXAMPLE. Let there be three places at distances from each other respectively of 50, 60, and 70 miles. Required the angle under which two roads must depart from that which is 60 and 70 distant from the other two, in the direction of these last. 60 and 70 will be the sides of a triangle adjacent the required angle, and 50 the side opposite; then Radius must be understood as a factor of the second member for the sake of homogeneity, since the quantity under the radical is the ratio of a surface to a surface, and therefore an abstract number. From the tables we find the angle to be 22° 12′ 28′′. 44° 24' 56' The other two angles may be found in a similar manner; the one is 67° 07′ 18′′, the other 78° 27′ 40′′. If R in the above formula should be made to pass (by squaring it) under the radical sign, it would be necessary to add twice 10 in order to effect the multiplication by this factor R2 before taking the square root. But as on the other hand twice 10 must be rejected for the arithmetical complements used, these two operations exactly counterbalance each other, and neither of them need be performed. By adding together the four logarithms, therefore, and dividing by 2, the same result will be obtained. The operation in the above example would be as follows: The best mode of proceeding in the solution of a plane triangle when three sides are given, is to prepare a blank form similar to that on p. 58, by ruling four columns, the first for the arguments, and each of the other three for the trigonometrical functions of those arguments necessary to be employed in the calculation of one of the three angles. Thus, a, b, c, denoting the given sides, s their sum, and A, B, C the required angles, To show how this form is filled up take the following example. There are other forms which have some advantages over the above, and which may be derived in an analogous manner. They are as follows. The student may write the blank forms for these formulas as an exercise. EXAMPLES. 1. Given in a plane triangle the three sides 120, 112 65, and 112, to find the three angles. 57° 27' Ans. 57° 58' 39' 64° 34' 21'' * Derived from (5) and (6) of Art. 72, and (1) of Art 73. A convenient form. By dividing the formula sin A√cos A by the formula cos A= cosa is found tan or at once dividing (3) by = sin A from which (4) above may be derived; (4). In a similar manner may be found cota= 2 sec a sec a +1 2 sec a and coseca = sec a -1 74. Before treating of the only remaining case in the solution of triangles, it will be convenient to demonstrate some additional general formulas which shall present certain important relations of the trigonometrical lines of two different arcs; which formulas are of frequent use in the higher analysis, are employed in the subsequent parts of the work, and will be immediately of service in deriving a formula for the last case of plane trigonometry which we have to consider. Add together equations (3) and (7) of (Art. 70) which express the values of sin (a+b) and sin (ab) and the resulting equation, cancelling the second terms of the second members which are similar with contrary signs, is subtracting the same equation, the second from the first, substituting in equation (1) the values of a+b, ab, a and b in terms of p and q, that equation becomes sin p+ sin q2 sin (p+q) cos (p − q) * (2) Which may be translated into ordinary language thus: the sum of the sines of two arcs is equal to twice the sine of half the sum into the cosine of half the difference of those arcs. By subtracting the latter of the same equations (3) and (7) of (Art. 69) from the former, and reducing similar terms, there results sin (a + b) — sin (a - b) — 2 sin b cos a (3) * R must be understood either as a divisor of the second member or multiplier of the first, because the sum of two lines cannot be equal to a rectangle. making the same substitutions as above in equation (1) this last equation or, the difference of the sines of two arcs is equal to twice the sine of half their difference into the cosine of half their sum.* which may be expressed in a proportion thus: the sum of the sines of any two arcs is to the difference of their sines as the tangent of half their sum is to the tangent of half their difference. |