1. B.1.8 – Derivatives of Primary Trig Functions Calculus - Santowski 12/1/2015 1 Calculus - Santowski

2. Background Skills 1. Graphs of sin(x), cos(x), tan(x) & their features 2. Evaluating trig functions at a given point (No calc) 3. Trig identities – Pythagorean, add/sub, double angle 4. Solving simple trig equations 12/1/2015 Calculus - Santowski 2

3. 12/1/2015 Calculus - Santowski 3 Fast Five 1. State the value of sin(  /4), tan(  /6), cos(  /3), sin(  /2), cos(3  /2) 2. Solve the equation sin(2x) - 1 = 0 3. Expand sin(x + h) 4. State the value of sin -1 (0.5), cos -1 (√3/2) 5. Explain how to find the value of the lim x->0 (cosx - 1)/x 6. Explain how to find the value of the lim x->0 sin(x)/x

4. 12/1/2015 Calculus - Santowski 4 Lesson Objectives 1. Review the derivatives of trigonometric functions algebraically and graphically 2. Differentiate equations involving trigonometric functions 3. Apply sinusoidal functions and their derivatives to questions involving tangent lines and function analysis

5. (A) Review - Derivatives The derivative of y = sin(x) is dy/dx = cos(x) 12/1/2015 Calculus - Santowski 5

6. Trig Derivatives Using the derivative of y = sin(x) and your fundamental basic trig knowledge, determine the derivatives of the other 5 trig functions (a) y = cos(x) (b) y = tan(x) (c) y = sec(x) (d) y = csc(x) (e) y = cot(x) 12/1/2015 Calculus - Santowski 6

7. Differentiating with sin(x) & cos(x) Differentiate the following y = cos(x 2 ) y = cos 2 (x) y = 3sin(2x) y = 6xsin(3x 2 ) Differentiate the following: 12/1/2015 Calculus - Santowski 7

8. Applications – Tangent Lines Find the equation of the tangent line to f(x) = xsin(2x) at the point x = π/4 What angle does the tangent line to the curve y = f(x) at the origin make with the x-axis if y is given by the equation 12/1/2015 Calculus - Santowski 8

9. Applications – Curve Analysis Find the maximum and minimum point(s) of the function f(x) = 2cosx + x on the interval (-π,π) Find the minimum and maximum point(s) of the function f(x) = xsinx + cosx on the interval (-π/4,π) Find the interval in which g(x) = sin(x) + cos(x) is increasing on xER 12/1/2015 Calculus - Santowski 9

10. Applications Given (a) for what values of a and b is g(x) differentiable at 2π/3 (b) using the values you found for a & b, sketch the graph of g(x) 12/1/2015 Calculus - Santowski 10

11. 12/1/2015 Calculus - Santowski 11 Internet Links Calculus I (Math 2413) - Derivatives - Derivatives of Trig Functions from Paul Dawkins Calculus I (Math 2413) - Derivatives - Derivatives of Trig Functions from Paul Dawkins Visual Calculus - Derivative of Trigonometric Functions from UTK Visual Calculus - Derivative of Trigonometric Functions from UTK Differentiation of Trigonometry Functions - Online Questions and Solutions from UC Davis Differentiation of Trigonometry Functions - Online Questions and Solutions from UC Davis The Derivative of the Sine from IEC - Applet

12. 12/1/2015 Calculus - Santowski 12 Homework Handout from Stewart, Calculus: A First Course, 1989, Chap 7.2, Q1&3 as needed, 4-7,9