Schaum's Outline of Theory and Problems of Beginning Calculus

208 GRAPHS AND DERIVATIVES OF SINE AND COSINE FUNCTIONS [CHAP. 2727.4 Prove Lemma 27.1.Consider the case when 8 > 0. Using the notation of Fig. 27-7,area (AOBC) < area (sector OBA) < area (AODA)t'IFig. 27-7Now1-- 1area (AOBC)2 2earea (sector OBA) = - x (area of circle)27t- e x (7t12) = - e27t 2= - (OC)(BC) = - cos 8 sin 8Furthermore, since AOBC and AODA are similar (by the AA-criterion),-DAOADA 1 - sin 8-=- or -=-or DA =-BCOC sin 8 COS 8 cos etinarea (AODA) = - (DAXOA) = --2 -- 2 cos 9and ( I) becomes, after dividing through by the positive quantity 4 sin 8,which is equivalent toe 1cos e < - < -sin 8 COS 8ALGEBRA If a and b are positive numbers, thena