§3.3 Derivatives of Trig Functions The student will learn about: Derivative formulas for trigonometric functions. 1

2 Some Preliminary Work #1 Reciprocals and inequalities. Remember if a, b and c are positive 0 < a < b < c, then If then 2

3 Some Preliminary Work #2 We need to establish the following important limit. ∆OMP is inside sector OMP which is inside ∆OMQ. Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ. 3 tan x Consider the drawing to the right. OAM (1, 0) P Q sin x x

4 Preliminary Work #2 Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ. and since and So by the “squeeze” theorem 4 tan x O AM (1, 0) P Q sin x x

5 Some Preliminary Work #3 5 We will also need to establish that = 0 ∙ 1 ∙ ½ = 0

Derivative of sin x Let f (x) = sin x and we will use the five step procedure to calculate the derivative. 1. f (x + h) = sin x cos h + cos x sin h 2. f (x) = sin x 3. f (x + h) – f (x) = sin x cos h + cos x sin h – sin x = sin x cos h – sin x + cos x sin h = sin x (cos h – 1) + cos x sin h = (sin x) · (0) + (cos x) · (1) = cos x

Derivative Formulas General Form - Chain Rule For u = u (x): Derivative Formulas for Sine and Cosine Basic Form 7

Examples 5 cos 5xa. y = sin 5x y’ = b. y = cos x 2 y’ =(2x)(- sin x 2 ) = - 2x sin x 2 8 c. y = (cos x) 2 y’ =(cos x)(- sin x) + (cos x) (-sin x = 2 sin x cos x

Examples Find the slope of the graph of f (x) = cos x at (π/4,  2/2), and sketch the tangent line to the graph at that point. We will use our graphing calculator to do this problem. slope 9

Examples Find the slope of the graph of f (x) = cos x at (π/4,  2/2), and sketch the tangent line to the graph at that point. Or we can use algebra to solve this problem. 10 y’ = - sin x y’ = - sin x so the m = y’ = - sin ( π/4) y’ = - sin x so the m = y’ = - sin ( π/4) = -√2/2. And the tangent goes through the point (π/4,  2/2), so using the point-slope form of a line

Summary General Form For u = u (x): Derivative Formulas for Sine and Cosine Basic Form 11

12 ASSIGNMENT §3.3; Page 53; 1 to 21 odd.