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§3.3 Derivatives of Trig Functions

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The student will learn about:

Derivative formulas for trigonometric functions.

Remember if a, b and c are positive

0 < a < b < c, then

If then

Reciprocals and inequalities.

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

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

We will also need to establish that

= 0 ∙ 1 ∙ ½ = 0

5

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):

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

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.

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):

ASSIGNMENT

§3.3; Page 53; 1 to 21 odd.