n(fx)' 10°. [l(a + bx + cx2+...)"]' = n(b + c • 2x + d • 3x2 + ...) 11°. (log. ) = M (†). [(298)] [[(298)]] BOOK SECOND. PLANE TRIGONOMETRY. SECTION FIRST. Trigonometrical Analysis. Construction. Describe the quadrant ABC; drop the perpendiculars BX, BY, upon the radii OA, OC; produce OB so as to intercept AT and CV, perpendiculars drawn through the extremities of OA, OC, in T and V; then: Y X Fig. 52. B a is the com Definition 1. The arcs AB, BC, are said to be complementary to each other. BC is the complement of AB and AB is the complement of BC; the arc 90° plement of a and a is the complement of 90° — a ; 45°+x and 45° ≈ are complementary arcs. Def. 2. The perpendicular BX is called the sine of the arc AB. Hence the sine of an arc is the perpendicular let fall from one extremity of the arc upon the diameter passing through the other extremity of the same arc. Def. 3. AT is the tangent of AB. Def. 4. OT is the secant of AB. Def. 5. BY (= OX) is the sine of BC or the cosine of AB. Def. 8. AX is the verst sine of AB. Note. The abbreviations of the titles above, either with or without the period, are employed as symbols of the quantities themselves. Thus, if a denote any arc less than a quadrant and b any arc not greater than 45°, the above definitions give TRIGONOMETRICAL ANALYSIS. sina = cos(90° — a), cosa = sin(90° — a) ; sin(90° — a) = cosa, cos(90° — a) = sina ; tan(90° — a); &c. sin(45°+b) = cos(45° - b), cos(45°+b) = sin(45°—b). cot(90° - a), cota tana = = seca = cosec(90° — a), coseca = sec(90° - a); &c. PROPOSITION I. The sum of the squares of the sine and cosine of an arc (305) is equal to the square of the radius, or to unity, when the radius is taken for the unit of the trigonometrical lines. We have or OB2 = (fig. 52.) when radius r = Cor. The sine is an increasing and the cosine is a de- (306) creasing function of the arc, or the sine BX increases from 0 to r as the arc increases from 0 to 90°, while the cosine OX decreases from r to 0 for the same increase of the arc. See (194) and observe that the sine BX is half the chord of double the arc AB. PROPOSITION II. The tangent of an arc is to the radius as the sine to the (307) cosine; or, the tangent is equal to the sine divided by the cosine, if the radius be taken for unity. The radius is a mean proportional between the tangent (308) and the cotangent of an arc; or, the tangent and cotangent of an arc are reciprocals of each other when r = 1. For, by similar triangles, we have = AT AO CO: CV, or tana: rr: cota; (fig. 52.) PROPOSITION IV. The square of the secant is equal to the sum of the (309) squares of the radius and the tangent. We have OT2 = OA2 + AT2, or sec2a = r2 + tan2a. (fig. 52.) The student may obtain other forms when wanted; as, for instance, the following: The secant is to the tangent as radius to the sine. Also (310) secant X cosine = r2 = 1. PROPOSITION V. AN INCREMENTAL VANISHING ARC is to be regarded as (311) a straight line perpendicular to the radius. Let AB be the arc in question; draw the tangents AT, BT, intersecting in T, and join OT; then will the triangles AOT, BOT, be equal, and OT will bisect the chord and arc in P and Q and be perpendicular to AB. From the similar triangles TAP, TOA, we have Fig. 53. B but QT = OT - OQ = (OA2 + AT2)* — OA, which reduces to [QT] = (OA2 — 02)*— OA = 0, when the arc AQB becomes = 0, since then AQ = †AQB = 0 and AT = tanAQ= 0; therefore the ultimate ratio of the vanishing quantities [AT], [AP], becomes But (113) the arc AQB is greater than the chord AB, and less AQB than the broken line ATB; .. the quotient is greater than AB AB AB or unity, and less than the arc AQB = 0; ATB which also becomes = 1, when DERIVATIVES OF SINE AND COSINE. 173 of the vanishing arc [AQB] to its vanishing = chord [AB] cannot be less than one, and cannot be greater than one, and therefore must be 1; which proves the proposition (300), observing that the arc is perpendicular to the radius (177). Scholium. It is not stated that the vanishing arc, when employed as an increment, merely may be regarded as a straight line perpendicular to the radius, but it is proved that it must be so regarded. On the other hand, it is to be observed that we do not affirm that the arc will ever actually become a straight line, or that it will not always exceed its chord in length, but that, for the purpose pointed out, it must be so regarded, in order to reduce it to zero, and thereby to eliminate it from the function under investigation. PROPOSITION VI. To find the derivatives of the sine and cosine regarded as functions of the arc. Denoting the arc by z, the sine by y, and the cosine by z, the functions may be represented by y = fx, z = 4x, which are the same as y = sinx, z=cost. Fig. 54. Attributing to the incremental vanishing arc h, (311), and to y, z, the corresponding increments, k,—i, (306), the similar triangles, whose homologous sides, taken in order, are k, h, -i; z, r = 1, y, give us The derivative of the sine regarded as a function of (312) its arc, is equal to the cosine, radius being unity. Again we have -=[]===—y; i. e., T The derivative of the cosine regarded as a function of (313) its arc, is equal to the sine taken minus. Proceeding to the 2d, 3d, 4th, &c., derivatives, observing that m (294) gives (ƒ)'=-(f') when = 1, we find, (312), (313), n |