Calculus: Analytic Geometry and Calculus, with Vectors

8.1 Trigonometric functions and their derivatives 441

Since 1 - cos 0 > 0 and 1 + cos 0 > 1 and sin 8 < 0, the last termcan
be overestimated by the inequality

(8.143) 0 < (1 - cos 0) < (1 - cos o)(1 +cos 0)

= 1 - cost 0 = sin2 0 < 0 sin 0.

Hence we can replace the last term in (8.142) by 0 sin 0 and divide by
sin 0 to obtain the first and hence the second of the significant inequalities

(8.144) (^1) _sin0<1+0,^11 <sin0<1
Dividing (8.143) by 0 and using (8.142) gives
(8.145) 0<1-Bos0<sin0<0.
The above inequalities have been proved to hold when 0 < 0 < 7r/2.
Since 101 = 0 when 0 > 0, (8.144) and (8.145) imply that the inequalities
(8.146)

• sin 0<1-cos0<iei
  1 101
  I0I
  hold when 0 < 0 < 7/2. Since the members of these inequalities are not
  changed when we replace 0 by -0, we conclude that they hold when
  181 < 7r/2 and 0 5 0. The desired formulas (8.14) follow from this and
  the sandwich theorem.
  To derive the formulas for derivatives of sines and cosines, we use the
  formulas
  sin (x + Ax) = sin x cos Ax + cos x sin Ax
  cos (x + Ax) = cos x cos Ox - sin x sin Ax
  to obtain
  sin (x + Ax) - sin x = - sin x(1 - cos Ax) + cos x sin Ax
  cos (x + Ax) - cos x = - cos x (1 - cos Ax) - sin x sin Ax.
  After dividing by Ax, taking limits as Ax -* 0 of the resulting difference
  quotients gives, with the aid of (8.14), the fundamental formulas
  (8.15) dsin x = cos x,
  d
  cos x = - sin x.