5.8 Inverse Trig Functions and Differentiation

Find the particular solution of the differential equation

that satisfies the initial condition.

' 0 (0) 4xyy e y

5.8 Inverse Trig Functionsand Differentiation

2

2

2

' 0 (0) 4

2

2 16 2

2 14

x

x

x

x

x

xy

yy e y

ydy e dx

ydy e dx

ye C

e

y e C C


Definition of Inverse Trig FunctionsFunction Domain Range

arcsin siny x y x 1,1 / 2, / 2

arccos cosy x y x 1,1 0,arctan tany x y x , / 2, / 2

cot coty arc x y x sec secy arc x y x csc cscy arc x y x

, 0,1x 0, , / 2y 1x / 2, / 2 , 0y

"The angle whose secanse t is "c arc x x

problemarcsecant

Function: y = arcsin x y = arccos x y = arctan x

Defined by: x = sin y x = cos y x = tan y

Domain:

Range:

Graph:

The Inverse Trig FCTNS

Function: y = arccot1 x y = arcsec1 x y = arccsc1 x

Defined b: x = cot y x = sec y x = csc y

Domain:

Range:

Graph:

The Inverse Trig FCTNS


1arcsin

2

DEFINITION

6

arccos0 2

arctan 3 3

arcsin(0.3) ????? Calculator

The restricted domain of / 2, / 2

this is the only solution.


Solve arctan 2 34

x

2 3 tan4

x

2 3 1x

2x


Given arcsin , where 0 / 2, find cos .y x y y sin y x

y

1 x

b2 2 21 x b 21b x

2cos 1y x

SohCahToa


5 /Gi 2ven arcsec , find tan .y y

y

5b

22 2 25 2 b

tan 1/ 2y

cos secADJ HYP

x xHYP ADJ

5sec

2y

1b


2 2

2 2

2

Derivatives of the Inverse Trig Functions

Let be a differentiable function of .

' 'arcsin arccos

1 1' '

arctan arccot1 1

' 'arcsec ar ccsc

1

u x

d u d uu u

dx dxu ud u d u

u udx dxu ud u d u

u udx dxu u u u

2 1

Problem


arcsin 2d

xdx

2

'arcsin

1

d uu

dx u

2

2

1 4x

arctan 3d

xdx

2

'arctan

1

d uu

dx u

2

3

1 9x


arcsind

xdx

1

2

/ 2 1

1 2 1

1/

2

1 2x

x x xx x

2

'arcsin

1

d uu

dx u

2 4

2

2 2( )

2

1

2

1

x

x x xe e e

e


2arcsec xde

dx

2

'arcsec

1

d uu

dx u u


2 arcsin 1Differentiate y x x x

2

'arcsin

1

d uu

dx u

1/ 22

22 2

2 2

2

2

2

2 1' 1 2

2

2 112

1

1

11

1

11 1

y x x x

x

xx

x x

x xx

x

A photographer is taking a picture of a 4' painting.

The camera lens is 1' below the lower edge of the painting.

How far should the camera be from the painting to

maximize the angle subtended by the camera lens?

Sol.

cot cot5 1

x x

cot cot

5

xarc arc x

2 21 / 25

1/ 5 1

1x

d

dx x

cot cot5

xarc arc x

2

'arccot

1

d uu

dx u

22

2

2

2

2 2

2

2

2 2

1/ 5 1

125 /

5 1

5 5 25

2

25

25

5 1

1

0

2 1

42

5

xx

x x

x x

x

x

x

x

x

Problem

2

2 2

4 5

25 1

x

x x

5x

is the location of a relative extrema.

'( 4) 0 '( 6) 0f f 5x yields a maximum value of .

2.236 feetx

Basic Differentiation Rules for Elementary Functions

HW 5.8/1,3,7,9,15,19,21,25,29,35,41,

45-55odd,59,67-70

False: arccos 1/ 23. / 33 1

9. arctan "The # whose tangent is "3 3

x

21

3 / 6x

secy x

1

2

1x in 0, , / 2y y secy arc x

1 1

Definitions

15. arcsec 1.269

1.269 sec

1/1.269 cos

arccos 1/1.269 0.6632

x

x

x

x

19. sin arctan 3/ 4

x

53

4

3

5

sec arcsin 4 / 5

x

54

3

5

3

25. sin ar csecy x

x2 1b x

1

2 1xy

x

because secant may be negative from 0 to

and sine is positive.

x

siny xsecy x

41. arcsin 2 arccos

2 sin arccos

1

3

2 1

2 1

x x

x x

x x

x x

x

1

1b x

x

41. arcsin 2 arccosx x

12x

2 221 2

1 2 1/

'07

3

J

x x

ake Gabl

x

e

x x

1

x

2 22

45. 3arccos / 2

1/ 2 1/ 2' 3 3

1 / 4 4 /

3

44

g x x

g xx x x

2

'arccos

1

d uu

dx u

2 2 2 2 2 2 2

47. arctan /

1/ 1/'

1 / /

g x x a

a ag x

x

a

xx aa a a

2

'arctan

1

d uu

dx u

2

2

2

2

2

2 2

arcsin 349.

3 /( 1 9 ) arcsin 3 1'

3 / 1 9 arcs

3 arcsin 3

in 3

1 9

1 9

xg x

x

x x xg x

x

x x x

x

x x x

x x

2

'arcsin

1

d uu

dx u

2

2

51. sin arccos

1' cos ar os

1

cc1

h t

t

t

t

h t tt

2

'arccos

1

d uu

dx u

2

'arctan

1

d uu

dx u

2

2 2

2 2

2 2

44

1 1 1 1 1 153. ln arctan ln arctan

2 2 1 4 1 2

1 1ln 1 ln 1 arctan

4 21 1 1 1 1

'4 1 1 2 1

1 11 1 1

4 21 1

1 1

2 1 2 1

1 1 1

1

1

2 1 1

x xy x x

x x

y x x x

yx x x

x x

x x

x x

x x

xx

4x

2

2 2 2

4 2

2

59. arcsec

1' 1 0

1

1 1 1 1

1 0

1 1 4 1 1 1 5

2 2

1 51.272

2

f x x x

f xx x

x x x x

x x

x

x

2

'arcsec

1

d uu

dx u u

Solve without a calculator:

3sin arctan

4

Quiz

53

4

sin 3/ 5

2

3 2

2

2

4 212)

3 21

20) ln 121

22) ln4

24) 1/2

126) ln 2

2

1 339) a. sin b. cos

2 2

x x

x x x

x C

x C

140) a. cot 2 tan

2

1b. sec 5 cos

5

3

3/ 22

2

2

1/

2 2

1

51) 1

53) ' sec1

55) ' arcsin

63) 6.93%

66)

168) ln

21

70) 3

x

x x

x

x

xy arc x

x x

y x

r

e C

e e C

e C

2

2

83) 3ln2

85) x

xy x C

y Ce

Chapter Review

89) a. Graph b. Greatest when 0, Least when 1

c. sin , / 2 / 2

y y

y x c x c

92) 7.79

Red Review for 5.6-5.8

2

2

83. Solve the differential Equation.

3

3l

3

n2

'

3

xy dy x dx

x x

dy

xy x

dx

C

x x

2

2

85. Solve the differential Equation.

' 2 0 2 2

2

ln

x

dy dyy xy xy xdx

dx y

dyxdx

y

y x

y Ce

C

2

2

2

2

2

89. ' 1

1

1

1

1

ar

Greatest 0

Least 1

csin

'arcsin

i

2

1

s n

y y

dyy

dxdy

dx

d u

y

y

y

y x C

dy dxy

y x C

udx u

x C

0

18,000

18,000

0.000038508

92.

0 30 30

30

18,000 30 15

1

2ln 0.5

0.00003850818,000

30

35,000 7.79

kh

kh

k

k

h

P h Ce

P Ce C

P h e

P e

e

P

k

P h e

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5.8 Inverse Trig Functions and Differentiation