sin

is not one-to-one

does not have an . inverse

y x

y

*** Our restricted domains will be the largest interval

(containing q

sin with re

uadrant I) in which is strictly monotoni

stricted domain in ,2 2

c.

x

f

x

,12

, 12

1,

2

1,2

1

12

2

x

y

siny x 1

1

y

x2

2

11

y

x

2

2

sin with

restricted domain

y x1siny x

(with the appropriate restricted domaiSine

is one-to-one and does have an in

n)

verse.

3

2

2 3

2

2

Restricted : /2 2

1

s

/

i

: 1

n x

D x x

R y y

1sin

: / 1 1

: /2 2

D x x

R y y

x

1sin is the unique in , such that sin2 2

y x y x

1,2

1,2

11

y

x

2

2

1siny x

1. . sin 1 sin 12 2

i e

,12

, 12

1

1

y

x2

2

sin with

restricted domain

y x

Solve for xOur restricted domain of sine

x

sin x

0 2 3 4 6 6 4 3 2

1 3

2

2

2

1

2 0

1

2

2

2

3

21

x

1sin x

1 3

2

2

2

1

2 0

1

2

2

2

3

21

2

3

4

6

0

6

4

3

2

,12

, 12

1,

2

1,2

1

12

2

x

ysiny x 1

1

y

x2

2

11

y

x

2

2

sin with

restricted domain

y x1siny x

1sin sin ?4

1 5sin

5Expla siin why n .

4 4

1 5in

4s sin

4

4

1 2

2sin

4

If our initial is not in the restricted domain of sine, we must find an

(within the restricted domain) for which sine has an equivalent output.

1cos is the unique in 0, such that cosy x y x

To compute the derivatives of the inverse trigonometric functions we simply need to simplify composite expressions, such as cos(sin−1 x) and tan(sec−1 x), by referring to the appropriate right triangle.

, 1

0,1 1,

1,0

1

12

2

x

ycosy x 1

1

y

x2

11

y

xcos with

restricted domain

y x1cosy x

cosine of "the whose sine is "xSimplify cos(sin−1 x) and tan(sin−1 x).

-1 2c 1os sin cosx x

-1

2tan sin

1tanx

x

x

2 2 1b x

1

2

1

1

1 1si

sin sin

1 1

coscos s 1in

n

dx

f x x f x g x x

xdx x

1

1

2

1 1cos cos

1 1

sinsin

1cos

1cos

f x x f x g x x

dx

dx x x

THEOREM 1 Derivatives of Arcsine and Arccosine

Derivative of an i1

'v rse'

n e g xf g x

CV

1 1

2 2

1 1sin , cos

1 1

d dx x

dx dxx x

I can't use the same ,

but I can use the same right :-)

Derivatives of Arcsine and Arccosine 1g x f x

QED

Right 's THUse and to find thM 2 . e..

1

2

1sin

1

dx

dx x

If f (x) = arcsin(x2)

1 ' ?

2f

1 2

4

2 1 1sin '

4 4

2 11 116

15

15

1

151

156

d xx f

dx x

or for McNeal...

1tan is the unique in , such that tan2 2

y x y x

1cot is the unique in 0, such that coty x y x

1sec is the unique in [0, ) ( , ] such that sec2 2

y x y x

secy x

x

y y

1secy x

x

1csc is the unique in [ ,0) (0, ] such that csc2 2

y x y x

2

2

1

1

2

2

x

111csc x cscf

y

cscy x

x

y

1cscy xx

THEOREM 2 Derivatives of Inverse Trigonometric Functions

1 12 2

1 1

2 2

1 1tan , cot

1 11 1

sec , csc1 1

d dx x

dx x dx xd d

x xdx dxx x x x

122

3

9 6

1 3tan

1 1 23 1

3

dx

dx x x x

Day 2

THEOREM 2 Derivatives of Inverse Trigonometric Functions

1 12 2

1 1

2 2

1 1tan , cot

1 11 1

sec , csc1 1

d dx x

dx x dx xd d

x xdx dxx x x x

1

20

0

20 0

1csc 1

1 1 1

1 1 21

1

3

x

x

x xx

ede

dx e e

e

e e

Sorry McNeal...

The formulas for the derivatives of the inverse trigonometric functions yield the following integration formulas.

Integral Formulas

1

2

12

1

2

sin1

tan1

sec1

dxx C

xdx

x Cxdx

x Cx x

In this list, we omit the integral formulas corresponding to the derivatives of cos−1 x, cot−1 x, and csc−1 x

... are nothing more than simple transformations

of the antiderivatives shown above.

1

2

1

2

1sin

1

sin1

dx

dx xdt

t Ct

We can use these formulas to express the inverse trigonometric functions as definite integrals. For example, because sin−1 0 = 0, we have:

1

20

sin for 1 11

x dtx x

t

Area model, in terms of .x

0C

11

0tan

40

4x

1

20

?1

dx

x

221

22

1

2

1sec 2 s

12 sec

11

ec 2

2

duu

u u

2 2u x du dx

1

2sec

1

dxx C

x x

Using Substitution1

21/ 2

?4 1

dx

x x

Because of our bounds,

the is not necessary.

4 4

3 3u x du dx

222 16 4

9 16 9 1 3 19 3

x xx

0 0

2 21 1

01

1

31443 1 1

1 1sin 0

4 4 2 8

du du

u u

u

Using Substitution

1

2sin

1

dxx C

x

0

23/ 4

?9 16

dx

x

21

dx

x

Suggested Problems 2 Days

Day 1: 5,7,17,21,27,29,33,37,41,45

Day 2: 59-107 EOO (Use integration techniques discussed

thus far in the class)