Drill

• Convert 105 degrees to radians

• Convert 5π/9 to radians

• What is the range of the equation y = 2 + 4cos3x?

• 7π/12

• 100 degrees

• [-2, 6]

Derivatives of Trigonometric Functions

Lesson 3.5

Objectives

• Students will be able to– use the rules for differentiating the six basic

trigonometric functions.

Find the derivative of the sine function.xy sin

hxfhxfxf

h

0lim'

H

xHxyH

sinsinlim'0

HxHxHxy

H

sinsincoscossinlim'0

HHxxHxy

H

sincossincossinlim'0

H

HxHxyH

sincos1cossinlim'0

Find the derivative of the sine function.xy sin

hxfhxfxf

h

0lim'

H

xHxyH

sinsinlim'0

HxHxHxy

H

sinsincoscossinlim'0

HHxxHxy

H

sincossincossinlim'0

H

HxHxyH

sincos1cossinlim'0

H

HxH

HxyHH

sincoslim1cossinlim'00

H

HxHHxy

HH

sinlimcos1coslimsin'00

1cos0sin' xxy

xy cos'

Find the derivative of the cosine function.xy cos

hxfhxfxf

h

0lim'

H

xHxyH

coscoslim'0

HxHxHxy

H

cossinsincoscoslim'0

HHxxHxy

H

sinsincoscoscoslim'0

H

HxHxyH

sinsin1coscoslim'0

Find the derivative of the cosine function.xy cos

H

HxHxyH

sinsin1coscoslim'0

H

HxH

HxyHH

sinsinlim1coscoslim'00

H

HxHHxy

HH

sinlimsin1coslimcos'00

1sin0cos' xxyxy sin'

Derivatives of Trigonometric Functions

xxdxd cossin

xxdxd sincos

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy cos3

33 coscos xdxdxx

dxdx

dxdy

23 3cossin xxxxdxdy

xxxxdxdy sincos3 32

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy

cos2sin

2cos2

cos2sinsincos2

x

xdxdxx

dxdx

dxdy

2cos2

sin0sincoscos2x

xxxxdxdy

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy

cos2sin

2cos2

sin0sincoscos2x

xxxxdxdy

2cos2

sinsincoscos2x

xxxxdxdy

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy

cos2sin

2cos2

sinsincoscos2x

xxxxdxdy

2

22

cos2sincoscos2

xxxx

dxdy

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy

cos2sin

2

22

cos2sincoscos2

xxxx

dxdy

2

22

cos2cos1coscos2

xxxx

dxdy

Remember that cos2 x + sin2 x = 1So sin x = 1 – cos 2x

Example 1 Differentiating with Sine and Cosine

Find the derivative.

xxy

cos2sin

2

22

cos2cos1coscos2

xxxx

dxdy

2cos21cos2

xx

dxdy

Homework, day #1

• Page 146: 1-3, 5, 7, 8, 10• On 13 – 16

Velocity is the 1st derivative Speed is the absolute value of velocity Acceleration is the 2nd derivative Look at the original function to determine

motion

Find the derivative of the tangent function.xy tan

xxy

cossin

x

xdxdxx

dxdx

y 2cos

cossinsincos'

x

xxxxy 2cossinsincoscos'

xxxy 2

22

cossincos'

Find the derivative of the tangent function.xy tan

xxxy 2

22

cossincos'

xy 2cos

1'

2

cos1'

xy

xy 2sec'

Derivatives of Trigonometric Functions

xxdxd cossin

xxdxd sincos

xxdxd 2sectan

Derivatives of Trigonometric Functions

xxdxd cossin

xxdxd sincos

xxdxd 2sectan xx

dxd 2csccot

xxxdxd tansecsec

xxxdxd cotcsccsc

More Examples with Trigonometric FunctionsFind the derivative of y.

xxy cot11sin

1sincot1cot11sin xdxdxx

dxdx

dxdy

xxxxdxdy coscot1csc1sin 2

xxx

xx

dxdy cos

sincos1

sin11sin 2

xxx

xxdxdy

sincoscos

sin1

sin1 2

2

xxxxxdxdy csccoscoscsccsc 22

xxxxxdxdy csccoscoscsccsc 22

xxxxxdxdy 22 csccoscsccoscsc

xxxxdxdy 22 csccos)cos1(csc

xxxxdxdy 22 csccos)(sincsc

xxxxdx

dy 22 csccos)(sinsin

1

xxxdxdy 2csccossin

More Examples with Trigonometric Functions

Find the derivative of y.

3

3

sec2tanxxxxy

23

3333

sec

sec2tan2tansec

xx

xxdxdxxxx

dxdxx

dxdy

23

23223

sec3tansec2tan6secsec

xxxxxxxxxxx

dxdy

52323

223

6secsec6sec

6secsec

xxxxxx

xxxx

5322

23

6tansec2tan3tansec

3tansec2tan

xxxxxxxx

xxxxx

)6tansec2tan3tan(sec6secsec6sec 532252323 xxxxxxxxxxxxxx

xxxxxxxxxxxx tansec2tan3tansecsecsec6sec 3222323

23

3222323

sectansec2tan3tansecsecsec6sec'

xxxxxxxxxxxxxxy

Whatta Jerk!Jerk is the derivative of acceleration. If a body’s position at time t is s(t), the body’s jerk at time t is

.3

3

dtsd

dtdatj

Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:

s1(t) = 3cos ts2(t) = 2sin t – cos t

Find the jerks of the bodies at time t.

tts cos31

tdtds

sin31 tsin3

tdt

sdcos32

12

velocity

acceleration

Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:

s1(t) = 3cos ts2(t) = 2sin t – cos t

Find the jerks of the bodies at time t.

tts cos31

tdtds

sin31 tsin3

tdt

sdcos32

12

velocity

acceleration

tdt

sdsin33

13

tsin3jerk

Example 2 A Couple of JerksTwo bodies moving in simple harmonic motion have the following position functions:

s1(t) = 3cos ts2(t) = 2sin t – cos t

Find the jerks of the bodies at time t.

ttts cossin22

ttttdtds

sincos2sincos21

ttttdt

sdcossin2cossin22

12

velocity

acceleration

ttttdt

sdsincos2sincos23

13

jerk

Homework, day #2

• Page 146: 4, 6, 9, 11, 12, 17-20, 22 28, 32