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7.7 Inverse Trigonometric

Functions

1M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Defining the Inverses

2M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

The six basic trigonometric functions are not one-to-one.

However we can restrict their domains to intervals on which

they are one-to-one.

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Defining the Inverses

3M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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The Domain and the Range for inverse trigonometric function

5M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

Function Domain Range

-1

y = cos x 1 1x 0 y

-1y = sin x

-1y = tan x

-1y = sec x

-1y = csc x

-1

y = cot x

1 1x

x

1x

2 2y

0 ,2

y y

0 y

2 2y

, 02 2

y y

1x

x

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Graphs for inverse trigonometric function

6M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Graphs for inverse trigonometric function

7M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Composition of Trigonometric Functions and Their Inverses

8M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

The Sine Function and Its Inverse

sin (sin-1x) = x for every x in the interval [-1, 1].

sin-1(sin x) = x for every x in the interval [-/2,/2].

The Cosine Function and Its Inverse

cos (cos-1x) = x for every x in the interval [-1, 1].

cos-1(cos x) = x for every x in the interval [0, ].

The Tangent Function and Its Inverse

tan (tan-1x) = x for every real number x

tan-1(tan x) = x for every x in the interval (-/2,/2).

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Example 1

9M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Example 2

10M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Example 2 cont.

11M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Example 3

12M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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Example 3 cont.

13M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

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The Derivative of y= sin-1 u

14M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

y = sin-1x is equivalent to sin y = x

Using implicit differentiation,

cos y y= 1

2

1

2

1cos

1 1

1

1If sin 1

1

yy

y xx

duy u y u

dxu

x1

y

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The Derivative of y= tan-1 u

15M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

x

1

21 x

1

2

2 2 2

tan tan tan

tan

sec . 1

1 1 1

sec 1 tan 1

y x x

d dy x

dx dx

dyy

dxdy

dx y y x

y

1

2

1tan

1

d duu

dx dxu

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Derivatives of inverse trig functions

17M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

1 1

2 2

1 1

2 2

1 1

2 2

1 1

sin cos 11 1

1 1tan cot

1 1

1 1sec csc 1

1 1

d du d du

u and u udx dx dx dxu u

d du d duu and u

dx dx dx dxu u

d du d duu and u u

dx dx dx dxu u u u

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Integration Formulas

18M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

The following formulas hold for any constant 0a

1 2 2

2 2

12 2

1

2 2

1. sin (Valid for )

12. tan (Valid for all )

13. sec (Valid for 0)

du uC u a

aa u

du u C ua u a a

du uC u a

a au u a

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Example 5

20M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

1

Find an equation for theline tangent to the graph

cot , 1y x at x

Solution:

1

2 21

3 3cot 1 1,4 4

1 1 1

21 1 1

3 11

4 2

x

dy dyslope

dx dxx

y x

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Example 6

21M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

1

Find tan

d

x xdx

Solution:

1/2 1122

1

1( ) tan

1 ( )

tan2(1 )

dyx x xdx x

x

xx

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Example 7

22M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

2

11

1

Find lim secx

x

x

Solution:

22

11 1

2

2

21

2

1

1.2

1 0 2 1lim lim1sec 0

1

lim . 11

lim 1

x x

x

x

x

x xx

x x

xx x

x

x

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Example 8

23M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

2

11

1

Find lim secx

x

x

Solution:

22

11 1

2

2

21

2

1

1.2

1 0 2 1lim lim1sec 0

1

lim . 11

lim 1

x x

x

x

x

x xx

x x

xx x

x

x

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Example 9

24M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

2

1

2 2

Evaluate

9

sin 33

dx

x

dx x

cx

Solution:

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Example 10

25M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

4

4 2 2

1 1 2

2

Evaluate

1

12

21 1 1

1 1sin

2 2

sin2 2

y dy

y

duy

y dy duy

y

du

let u y du y dy

u u

u c

d

y c

y y

Solution:

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Example 11

26M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

/4

1

4Evaluate

21 ln

4 4 1 14 4tan

22 2 11 ln 1

1

4tan ln

41 1 1 1 144tan ln 4 tan ln 4tan ln1 4tan 0 4ta

1

n

ln

4 41

edt

I

t t

dt t dudu u c

ut t t u

t c

eI t

let u t du dt dt t du

t

e

Solution:

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Example 12

27M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

2

22 2

22

1 1

2

Evaluate2 4 3

4 3 4 4 4 3 2 1

2 4 3 2 2 1

sec se 2

2

c1

dx

x x x

x x x x x

dx dxI

x x x

let u

x x

duI u c x cu

d x

u

x u d

Solution:

E l 13

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Example 13

28M.Hamouri, Y.Shehadeh Calculus II 7.7 PPU

2

22 2

1

22

Evaluate 6 10

6 10 6 9 9 10 3 1

tan 36 10 3 1

dy

y y

y y y y y

dy dyy c

y y y

Solution:

E l 14

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Example 14

29M Hamouri Y Shehadeh Calculus II 7 7 PPU

1

2

2

1

1

1sec

cos sec

Evaluate 1

cos sin sin sec

1

x

dx Ix x

I

let

u du u c x c

u x du dxx x

Solution: