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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. 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 -0.2 1 2 3x 4 5 0 -0.2 6 -0.4 -0.4 -0.6 -0.6 -0.8 -0.8 -1 -1 y sin x, 0 t x t 2= 1 2 3x 4 5 6 y cos x, 0 t x t 2= 1 0.5 -6 -4 -2 0 2 x 4 6 -0.5 -1 — y sin x, -.-. y cos x, " 2= t x t 2= Relations (identities): a. sin x 1 i. tan x cos ii. cot x cos x 1 iii. sec x cos iv. csc x 1 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 using the Quotient Rule. b. More commonly used identities: ii. sec 2 x " tan 2 x 1 i. sin 2 x cos 2 x 1 iii. sin2x 2 sin x cos x iv. cos2x cos 2 x " sin 2 x 2 cos 2 x " 1 1 " 2 sin 2 x c. Sum-difference formulas: i. sina o b sina cosb o sinb cosa ii. cosa o b cosa cosb # sina sinb Values of Trigonometric Functions at Special Angles: 1 x in radians (degrees) ( sin x cos x tan x cot x csc x sec x 00 0 1 0 . . 1 = 30 ( 6 1 2 3 2 1 3 3 2 2 3 = 45 ( 4 1 2 1 1 2 2 = 60 ( 3 = 90 ( 2 3 2 1 2 3 1 3 2 3 2 1 0 . 0 1 . 1 2 2. Derivatives of sin x and cos x : d sin x cos x d cos x " sin x and dx dx Two facts are used in derivation: let 2 be an angle in radians and lim 1 " cos 2 0 lim sin 2 1 2 2 2v0 2v0 Derivative of sin x : Let fx sin x. Then U fx h " fx sinx h " sin x lim lim sin x cos h sinh cos x " sin x f x lim hv0 hv0 hv0 h h h sin xcos h " 1 sin h cos x sin h lim sin x lim cos h " 1 cos x lim hv0 hv0 hv0 h h h sin x 0 cos x 1 cos x In the derivation of the derivative of cos x, you will need to use the identity: cosx 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 0cos x " 1 " sin x d sec x d sin x 1 1 sin2x cos x cos x tan x sec x cos x dx dx cos 2 x cos x Example Let fx sin x and gx 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 U Example Compute f x where 2 b. fx 2 sin x cos x a. fx x sin x c. fx sec 2 x " tan 2 x d. fx 2 sec 2 x a. 2 U f x 2x sin x "2x cos x 2x " x 2 cot x csc x sin x sin x b. U f x 2¡cos 2 x sin x" sin x ¢ 2 cos 2 x " sin 2 x 2 cos2x c. U f x d sec 2 x " tan 2 x d 1 0 dx dx d. U f x 2 d sec x sec x 2 tan x sec x sec x sec xtan x sec x dx 4 tan x sec 2 x Example Find the equation of the tangent line to the curve y x 2 cos x at a = . 3 U = = = 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 U f x 2x cos x x 2 " sin x 2x cos x " x 2 sin x f U = 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