Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

3.5Derivatives of Trigonometric

Functions

Slide 3- 2 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Quick Review

1. Convert 135 degrees to radians.

2. Convert 1.7 radians to degrees.

3. Find the exact value of sin without a calculator.3

4. State the domain and the range of the cosine function.

5. State the domain

and the range of the tangent function.

Slide 3- 3 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Quick Review

2

3 2

6. If sin 1, what is cos ?7. If tan 1, what are two possible values of sin ?

1 cos sin8. Verify the identity: .1 cos

9. Find an equation of the line tangent to the curve

2 7 10 at the point 3,

a aa a

h hh h h

y x x

3 2

1 .

10. A particle moves along a line with velocity

2 7 10 for time 0. Find the accelerationof the particle at 3.

v t t tt

Slide 3- 4 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Quick Review Solutions

1. Convert 135 degrees to radians.

2. Convert 1.7 radians to degrees.

3. Find the exact va

3 2.3564

97.403

32

lue of sin without a calculator.3

4. State the domain and the range of the cosine function.D

5. State the domain and the range of the tangent functomain: all reals Range: [-1,1]

Domain: odd integer Range:

ion.

all reals2

kx k

Slide 3- 5 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Quick Review Solutions

2 2

2

6. If sin 1, what is cos ?

7. If tan 1, what are two possible values of sin ?

1 cos sin8. Verify the identi

012

1 cosMultiply by and use the identity

ty: .1 cos

9. Find an equa

1 cos sin1 os

ic

t

a a

a a

h hh h h

h h hh

3 2

3 2

on of the line tangent to the curve

2 7 10 at the point 3,1 .

10. A particle moves along a line with velocity

2 7 10 for time 0. Find the accelerationof t

12 35

1he particle at 3 2.

y x x

v t t t

x

t

y

Slide 3- 6 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

What you’ll learn about

Derivative of the Sine Function Derivative of the Cosine Function Simple Harmonic Motion Jerk Derivatives of Other Basic Trigonometric Functions

… and why

The derivatives of sines and cosines play a key role in

describing periodic change.

Slide 3- 7 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Derivative of the Sine Function

The derivative of the sine is the cosine.

sin cosd x xdx

Slide 3- 8 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Derivative of the Cosine Function

The derivative of the cosine is the negative of the sine.

cos sind x xdx

Slide 3- 9 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Example Finding the Derivative of the Sine and Cosine Functions

sinFind the derivative of .

cos 2x

x

2

2

2 2

2

2 2

2

2

quotient rule

2 2sin cos 1

cos 2 sin sin cos 2

cos 2

cos 2 cos sin sin

cos 2

cos 2cos sincos 2

sin cos 2cos

cos 2

1 2coscos 2

x x

d dx x x xdy dx dxdx x

x x x x

x

x x xx

x x x

x

xx

Slide 3- 10 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Simple Harmonic Motion

The motion of a weight bobbing up and down on the end of a string is an example of simple harmonic motion.

Slide 3- 11 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Example Simple Harmonic Motion

A weight hanging from a spring bobs up and down with positionfunction 3sin in meters, in seconds . What are its velocity

and acceleration at time ?

s t s t

t

3sin

3cos

3sin

s tdsv tdtdva tdt

Slide 3- 12 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Jerk

3

3

is the derivative of acceleration. If a body's position attime is

.

t

da d sj tdt dt

Jerk

Slide 3- 13 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Derivative of the Other Basic Trigonometric Functions

2

2

tan sec

cot csc

sec sec tan

csc csc cot

d x xdxd x xdxd x x xdxd x x xdx

Slide 3- 14 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Example Derivative of the Other Basic Trigonometric Functions

Find the equation of a line tangent to cos at 1.y x x x

cos

cos sin cos 1

Evaluate when 11 .8414709848 .5403023059 .3011686789

1 .8414709848 .5403023059 .3011686789

When 1, 1 cos1 .5403023059

The equation of the tangent line is.54030

y x xdm x x x x xdx

m xm

m

x y

y

23059 .3011686789 1

.3011686789 .3011686789 .5403023059.3011686789 .8414709848

After rounding the equation is

x

y xy x

y = .3012x .841

Slide 3- 15 Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Prentice Hall

Example Derivative of the Other Basic Trigonometric Functions

cosy x x

.3012 .841y x