3.4 Derivatives of Trigonometric Functions

Review Trigonometric Functions – There are reference pages in front and back of textbook

Radian measure is necessary for all the following!!

We first need some special Limits necessary to find derivative of  sin(x).

Recall that we “hypothesized” the derivative of sin x in Section 3.2 from the graph, interpreting the derivative as the slope of tangent lines.

Will use  Sandwich Theorem to  prove

And as a result

Transparency

Unit Circle

DAOD  and  DBOC  and sector AOC

|AC|  = h    (definition of radian)

Let's find Area DAOD,  Area Sector A0C,  Area DBOC

Area AOD

Area Sector AOC

Area DBOC

Area DAOD  <  Area Sector AOC   <   Area DBOC

and divide by sin h

If positive values, reciprocating also flips inequalities:

Or

FINALLY!!! We can use the Sandwich Theorem for Limits

So

To Prove:

Must multiply by (1+cos q)/(1+cos q) and simplify and factor

Derivative of sin x

Use the definition of the derivative

Similar for Derivative of cos x

Let's use quotient rule to find derivative of   tan x

Could also use quotient rule for cot x, sec x, and csc x  (2 of these are homework problems!)

Derivative Rules for Trig Functions - Page 152

Examples:

#24  Find the equation of tangent line at (0,1) for

#40  Find the limit

Assignment:

3.4; pg. 154;  1, 5-19 odd, 23, 31, 33, 39, 41