B.1.8 – Derivatives of Primary Trig

FunctionsCalculus - Santowski

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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

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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 limx->0 (cosx - 1)/x

• 6. Explain how to find the value of the limx->0 sin(x)/x

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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

(A) Review - Derivatives

• The derivative of

y = sin(x) is

dy/dx = cos(x)

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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)

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Differentiating with sin(x) & cos(x)

• Differentiate the following

• y = cos(x2)

• y = cos2(x)

• y = 3sin(2x)

• y = 6xsin(3x2)

• Differentiate the following:

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y(t) = 1+ cos t + sin2 t

y(t) =x 2

2 − cos πx( )

f (y) = y 2 cos 3y 3( )

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

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€

y =1

3sin3x

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

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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)

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€

g(x) =sin x 0 ≤ x ≤

2π

3

ax + b2π

3< x ≤ 2π

⎧

⎨ ⎪

⎩ ⎪

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Internet Links

• Calculus I (Math 2413) - Derivatives - Derivatives of Trig Functions from Paul Dawkins

• Visual Calculus - Derivative of Trigonometric Functions from UTK

• Differentiation of Trigonometry Functions - Online Questions and Solutions from UC Davis

• The Derivative of the Sine from IEC - Applet

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Homework

• Handout from Stewart, Calculus: A First Course, 1989, Chap 7.2, Q1&3 as needed, 4-7,9