Schaum's Outline of Theory and Problems of Beginning Calculus

CHAP. 281 THE TANGENT AND OTHER TRIGONOMETRIC FUNCTIONS 215Tkurem 28.1: The tangent and cotangent functions are odd functions that are periodic, of period A.That they are odd follows from Theorem 26.3,sin (- x) -sin xtan (-x) = ----- -tan xcos (-x) cos x1cot (-x) = --- 1 - -cot xtan (-x) - -tan xThe periodicity of period ~t follows from Problem 26.15(c) and (4,sin (x + z)--- -sin xtan (x + A) = - tan xcos (x + It) - -cos xTheorem 28.2 (Deriuatiues): D,(tan x) = sec2 xD,(cot x) = -csc2 x&(sec x) = tan x sec xD,(csc x) = -cot x csc xFor the proofs, see Problem 28.1.EXAMPLE From Theorem 28.2 and the power chain rule,D,2(tan x) = D,((sec x)’) = 2(sec x)D,(sec x)= 2(sec x)(tan x sec x) = 2 tan x sec’ xNow in (0, n/2), tan x > 0 (since both sin x and cos x are positive), making D:(tan x) > 0. Thus (Theorem 23.1), thegraph of y = tan x is concave upward on (0, 42). Knowing this, we can easily sketch the graph on (0, n/2), andhence everywhere (see Fig. 28-1).IIIIIIIIIII1‘ IIIFig. 28-1