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§3.3 Derivatives of Trig Functions. The student will learn about:. Derivative formulas for trigonometric functions. Some Preliminary Work #1. Remember if a, b and c are positive 0 < a < b < c, then. If then. Reciprocals and inequalities. 2. Some Preliminary Work #2. Q.
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§3.3 Derivatives of Trig Functions The student will learn about: Derivative formulas for trigonometric functions.
Some Preliminary Work #1 Remember if a, b and c are positive 0 < a < b < c, then If then Reciprocals and inequalities. 2
Some Preliminary Work #2 Q P Consider the drawing to the right. tan x sin x x O A M (1, 0) 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
Preliminary Work #2 Q P tan x sin x x O A M (1, 0) Area of ∆OMP ≤ area sector OMP ≤ area of ∆OMQ. and and since So by the “squeeze” theorem 4
Some Preliminary Work #3 We will also need to establish that = 0 ∙ 1 ∙ ½ = 0 5
Derivative of sin x Let f (x) = sin x and we will use the five step procedure to calculate the derivative. 2. f (x) = sin x 1. f (x + h) = sin x cos h + cos x sin h 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 for Sine and Cosine Basic Form General Form - Chain Rule For u = u (x): Derivative Formulas
Examples a. y = sin 5x y’ = 5 cos 5x b. y = cos x 2 y’ = (2x)(- sin x2) = - 2x sin x 2 c. y = (cos x) 2 y’ = (cos x)(- sin x) + (cos x) (-sin x = 2 sin x cos x
slope 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.
Examples And the tangent goes through the point (π/4, 2/2), so using the point-slope form of a line 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. 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. 10
Derivative Formulas for Sine and Cosine Basic Form General Form For u = u (x): Summary
ASSIGNMENT §3.3; Page 53; 1 to 21 odd.