Writing 90° for u in [22], sin (90° + v) = sin 90° cos v + cos 90° sin v 18. sin (u+v) sin (u — v) = (sin u + sin v) (sin u [22] × [26], sin (u + v) sin (u — v) = sin2 u cos2v - cos2 u sin2 v. =(sinu+sin v)(sinu—sin v). - sin v). 19. cos (u + v) cos (u — v) = (cos u + sin v) (cos u sin 3 u = sin (2 u+u) = [sin 2 u] cos u + sin u [cos 2 u] = [2 sin u cos u] cos u + sin u [cos2 u = = 2 sin u cos2 u + sin u cos2 u sin2 u] - sin3 u = 2 sin u (1 − sin2 u) + sin u (1 − sin2 u) — sin3 u - 4 sin3 u. - 3 cos u. (cos u — cos v)2 = (2 sin (u — v))2. a+b = tan-1a, v= tan-1b, then tan u = a, tan v = b. Substitut ing in (a), and we get the given relation. NOTE. - The following six formulas serve to change sums to products, and thus adapt them to logarithmic computation. [Use 4 u = 2u + 2 u in [23].] 50. Show that sin 4 u = 4 sin u cos u - 8 sin3 u cos u. 51. Show that cos 5 u = 5 cos u - 20 cos3 u +16 cos3 u. CHAPTER IV OBLIQUE TRIANGLES RELATIONS OF SIDES AND ANGLES 95. Notation. The angles will be denoted by A, B, C, and the opposite sides, respectively, by a, b, c. Let the angle, say A, be acute (Fig. 45), or each figure draw BD 1 AC, and let y = BD. ures [15], AD+DC= = obtuse (Fig. 46). In Then in both fig AC, or DC = b — AD. ... DC2 = b2 + AD2 – 2b × AD. Adding y2, DC2 + y2 = b2 + AD2 + y2 − 2 b × AD. But DC2 + y2 = a2, AD2 + y2 = c2, and since AD ÷ AB = cos CAB, AD c cos A. = By drawing perpendiculars from the vertices C and A to the opposite sides, we obtain, in a similar way: The relation expressed by [51], [52], [53], is often called the Law of Cosines, of which the following is the Translation: The square of any side of any triangle is equal to the sum of the squares of the other two sides, minus twice the product of these sides into the cosine of their included angle. While all other trigonometric relations of the sides and angles of plane triangles can be derived from the law of cosines, it is often more convenient to deduce them by independent methods. Since 97. Deduction of formulas by cyclical changes of the letters. a and A, b and B, or c and C, stand for any side of a triangle and the opposite angle, from any formula expressing a general relation between these parts another formula may be deduced by changing the letters in cyclical order. Thus, in [51] by changing the a to b, the b to c, the c to a, and the A to B, we obtain [52]; and in [52] by making the same or similar changes we obtain [53]. 98. Relation of the three angles. By Geometry, A + B + C = 180°, or C 180° - (A + B). This relation is called the Law of Angles. by functions thus: '= cos [180° — (A + B)] = It may be expressed cos C 99. Relation of the sides to their opposite angles. From Figs. 45, 46, we have This relation may be deduced from [51] and [52] by eliminating may be done thus: |