Trigonometric functions 3.4 Derivatives of the Trigonometric functions Rita Korsunsky

1 - cos x 0 < sin x < x 0 < < x P (cos x , sin x) x 1 P (cos x , sin x) In the Unit circle (Radius=1): P(cos x, sin x) sin x 1-cos x cos x 0 < sin x < x 1 - cos x 0 < < x By Sandwich Theorem :

Let x be any number between 0 and . Proof Let x be any number between 0 and . Area ∆OMP = OM • MP = cos x sin x Area ∆OUQ = • 1 • UQ = C u v O M x P cos x sin x U (1, 0) tan x Q

Trigonometric Limits

Example 1

Derivatives of Trigonometric Functions

PROOF

PROOF

PROOF

PROOF

Example 2 Solution By quotient rule

Example 3. Find g'(x) if g(x) = sec x tan x Solution: By product rule

Example 4. Find the slopes of the tangent lines to the graph of y=sin x, at the points with x coordinates (b) For what values of x is the tangent line horizontal? Solution The slope of the tangent line at (x,y) is (b) A tangent line is horizontal if its slope is zero. y ' = 0; that is, cos x = 0

Example 5 Find an equation of the normal line to the graph of y = tan x at point P(/4, 1) Solution slope of the normal line is Equation of the normal line is: or

Example 6 Find the first eight derivatives of f (x) = sin x Solution Since f (x) = sin x, it follows that if we continue differentiating, the same pattern repeats; that is