. Differential and integral calculus. - / = ^ -— ak. Vi - ** V(i-^)3 .l{x + df ! x + a 34. y =V/- ~- dy = (x — 2 a) V > s„dx. \ x — a y v V (# — «/ THE TRIGONOMETRIC FUNCTIONS. 35. 7^ differential of the sine of an angle is equal to the cosine of the angle into the differential of ^ ^^ ? the angle. Let POC be any angle gen-erated by the line OP, taken asthe linear unit, revolving upwardabout O as an axis; then, incircular measure, Length of PC = u — measure of POC. If length u becomes a uniformly changing variable at the in-stant the generating point reaches the position P, then, § 18, PT =
. Differential and integral calculus. - / = ^ -— ak. Vi - ** V(i-^)3 .l{x + df ! x + a 34. y =V/- ~- dy = (x — 2 a) V > s„dx. \ x — a y v V (# — «/ THE TRIGONOMETRIC FUNCTIONS. 35. 7^ differential of the sine of an angle is equal to the cosine of the angle into the differential of ^ ^^ ? the angle. Let POC be any angle gen-erated by the line OP, taken asthe linear unit, revolving upwardabout O as an axis; then, incircular measure, Length of PC = u — measure of POC. If length u becomes a uniformly changing variable at the in-stant the generating point reaches the position P, then, § 18, PT = du and DT = d sin u (since AP = sin u). From the right triangle DTP, we have, DT= PT cos DTP,, d (sin a) = cos a du (15). 36 Differential Calculus 36. The differential of the cosine of an angle is equal to minusthe sine of the angle into the differential of the angle. From the right triangle DTP, Fig. 4, we have DP=PTsmDTP,, — d cos u = sin u du,or d (cos u) = — sin udu (16) since OA = cos u and OA is a decreasing variable, § (8), , thus: let u = u in equa. (15), then //sin ( u\ = cos ( u\dl u\f , d (cos u) — — sin & dfo. 37. The differential of the tangent of an angle is equal to thtsquare of the secant of the angle into the differential of the angle. To prove d (tan u) = sec2 u du. From trigonometry, sin u tan u = . cos u Differentiating, , . cos^ (cos udu) — sin u (— sin udu) //(tan u) = ^ L— 1 L cosz u (cos2 u -|- sin2 u) ducos2 u 1 du, cos ^d(tan u)= sec* udu * . (17) Differentiation 37 38. The differential of the cotangent of an angle is equal to minusthe square of the cosecant into the differential of the angle. To prove d (cot //) = — esc2 u du. We know th
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectcalculu, bookyear1918
