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2.5 Derivatives of Trigonometric Functions 1. Six Trigonometric Functions and Identities: Six trigonometric functions: sin x, cos x, tan x, cot x, sec x, csc x, x is in radians Recall: Functions sin x and cos x are periodic functions with period 2!. They are continuous everywhere and graphically they are also differentiable everywhere. Their derivatives exist everywhere. -10 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 -0.2 -5 5 x 10 -10 -5 0 -0.2 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 5 x 10 y ! cos x y ! sin x Relations: 1 sin x cot x ! cos x ! 1 sec x ! cos csc x ! 1 tan x ! cos x x tan x sin x sin x So, if we know the derivatives of sin x and cos x, then we can derive the derivatives of tan x, cot x, sec x and csc x by the Quotient Rule. 2. Derivatives of sin x and cos x : d !sin x" ! cos x dx Recall: Two facts - " in radians lim sin " ! 1 " ""0 d !cos x" ! ! sin x dx and and lim 1 ! cos " ! 0 " ""0 1.5 0.4 1 0.2 0.5 -1.5 -1 -0.5 0 0.5 x1 -0.5 1.5 -0.4 -0.2 0 0.2 x -0.2 -1 -1.5 – y ! sin x, -.-. y ! x 1 -0.4 – y ! sin x, -.-. y ! x 0.4 1.5 0.4 1 0.2 0.5 -1.5 -1 -0.5 0 -0.5 0.5 x1 1.5 -0.4 -0.2 0 0.2 x 0.4 -0.2 -1 -1.5 -0.4 – y ! tan x, -.-. y ! x – y ! tan x, -.-. y ! x Proof (1) Graphically, sin x " x, and tan x # x. So, sin x " x " tan x. sin x sin x x sin x " x iff sin x " 1; x " tan x iff x " cos x iff cos x " x Hence, x cos x " sin x " 1. Because lim x"0 cos x ! 1, and lim x"0 1 ! 1, by the Squeeze Theorem we x lim sin x ! 1. x"0 (2) Recall: sin 2 " # cos 2 " ! 1, or 1 ! cos 2 " ! sin 2 " !1 ! cos 2 "" !1 ! cos "" !1 # cos "" lim 1 ! cos " ! lim ! lim " " ""0 ""0 ""0 "!1 # cos "" !1 # cos "" 2 sin " 1 ! lim ! lim sin " sin " ! !1"!0" 1 ! 0 " ""0 "!1 # cos "" ""0 2 !1 # cos "" Derivative of sin x : Let f!x" ! sin x. Then $ f!x # h" ! f!x" sin!x # h" ! sin x ! lim ! lim sin x cos h # sinh cos x ! sin x f !x" ! lim h"0 h"0 h"0 h h h sin x!cos h ! 1" # sin h cos x sin h ! lim ! sin x lim cos h ! 1 # cos x lim h"0 h"0 h"0 h h h ! sin x !0" # cos x !1" ! cos x In the derivation of the derivative of cos x, you may need to use the identity: cos!x # h" ! cos x cos h ! sin x sin h. 3. Derivatives of tan x, cot x, sec x, and csc x : d !cot x" ! ! csc 2 x, d !sec x" ! tan x sec x, d !csc x" ! ! cot x csc x d !tan x" ! sec 2 x, dx dx dx dx Derivations: Use the derivatives of sin x and cos x and the Quotient Rule d !tan x" ! d sin x ! cos x cos x ! sin x!! sin x" ! cos 2 x # sin 2 x ! 1 ! sec 2 x dx cos x dx cos 2 x cos 2 x cos 2 x 0!cos x" ! !1"!! sin x" d !sec x" ! d sin x 1 1 ! ! sin2x ! cos x cos x ! tan x sec x dx dx cos x cos 2 x cos x Example Let f!x" ! sin x and g!x" ! cos x. Find f !5" !x", f !2003" !x", g !5" !x" and g !2003" !x". 2 n f !n" !x" g !n" !x" 0 sin x cos x 1 cos x ! sin x 2 ! sin x ! cos x 3 ! cos x sin x 4 sin x cos x 5 cos x ! sin x 2003 ! cos x sin x $ Example Find f !x" where 2 a. f!x" ! x sin x c. f!x" ! sec 2 x ! tan 2 x b. f!x" ! 2 sin x cos x d. f!x" ! 2 sec 2 x a. 2 $ f !x" ! 2x sin x !2x cos x ! 2x ! x 2 cot x csc x sin x sin x b. $ f !x" ! 2#cos 2 x # sin x!! sin x"$ ! 2 cos 2 x ! sin 2 x ! 2 cos!2x" c. $ f !x" ! d !1" ! 0 dx d. $ f !x" ! 2 d !sec x sec x" ! 2#!tan x sec x" sec x # sec x!tan x sec x"$ ! 4 tan x sec 2 x dx Example Find the equation of the tangent line to the curve y ! x 2 cos x at a ! ! . 3 $ ! ! ! the equation of the tangent line: y ! f x! !f 3 3 3 2 2 1 ! !2 f ! ! ! cos ! ! ! 18 3 3 3 3 2 $ f !x" ! 2x cos x # x 2 !! sin x" ! 2x cos x ! x 2 sin x f $ ! 3 ! !2" ! 3 cos ! 3 ! ! 3 2 sin ! 3 2 the equation of the tangent line: y ! ! ! 18 3 ! ! 3 ! 3 ! ! ! 3 ! 3 2 3 2 2 3 2 x! ! 3