aim: differentiate inverse trig functions course: calculus do now: aim: how do we differentiate...

Aim: Differentiate Inverse Trig Functions Course: Calculus

1.5

1

0.5

-0.5

-1

-1.5

-2

-2 -1 1 2

f x = sin x

Do Now:

Aim: How do we differentiate Inverse Trig functions?

Does y = sin x have an inverse?

arcsin sin

: 1 1

:2 2

y x iff y x

Domain x

Range y

yes, but only in restricted domain.


Definition of Inverse Trig Functions

Function Domain Range

arcsin sin -1 1 -2 2

arccos cos -1 1 0

arctan tan - -2 2

arccot

y x iff y x x y

y x iff y x x y

y x iff y x x y

y x

cot - 0

arcsec sec 1 0 ,2

arccsc csc 1 - , 02 2

iff y x x y

y x iff y x x y y

y x iff y x x y y

arcsinx sin-1 xalternate notation

‘the angle whose sin is . . .’


Graphs of Inverse Trig Functions

1

-1

-2

-2 2

g x = sin-1 x 3

2

1

-2 2

g x = cos-1 x

4

2

-2

-4

g x = tan-1 x Domain

-1 1

Range

-2 2

x

y

Domain

-1 1

Range

0

x

y

Domain Range

- -2 2

x y

2

2


Graphs of Inverse Trig Functions

2

1

-1

-2

-2 2

y = arccsc x 3

2

1

-1

-2

-2 2

y = arcsec x

3

2

1

-1

-2

-3

-4 -2 2 4

y = arctan x

Domain

-

Range

0

x

y

Domain

1

Range

0 ,2

x

y y

Domain

1

Range

- , 02 2

x

y y

2

2

2


Evaluating Inverse Trig Functions

1Evaluate: arcsin

2

‘the angle whose sin is -1/2’

sin is negative in QIII and IV

arcsin sin -1 1 -2 2

y x iff y x x y

however, restricted domain

reference angle of /6

angle y is - /6 0º 30º 45º 60º 90º

sin 0

cos 1

tan 0

1

2

1

2

1

2

2

2

2

3

2

3

2

3

3

3

1

0

UND.



Evaluate: arccos0

‘the angle whose cos is 0’


angle y is /2

arccos iff cos -1 1 0y x y x x y however, restricted domain

0º 30º 45º 60º 90º

sin 0

cos 1

tan 0

1

2

1

2

1

2

2

2

2

3

2

3

2

3

3

3

1

0

UND.



Evaluate: arctan 3


angle y is /3

‘the angle whose tan is ’3

however, restricted domain

arctan iff tan - -2 2

y x y x x y

0º 30º 45º 60º 90º

sin 0

cos 1

tan 0

1

2

1

2

1

2

2

2

2

3

2

3

2

3

3

3

1

0

UND.


Using Inverses to Solve an Equation

Solve: arctan 2 34

x

tan arctan 2 3 tan4

x

take tan of both sides

2 3 1x inverse property

x = 2 solve for x


Using Right Triangles

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

Since y = arcsin x, then sin y = x

oppositexy

1

21 x

cos cos(arcsin )y x 2.1

.

adjx

hyp



5 . 1tan tan sec

2 . 2

oppy arc

adj

Given sec 5 / 2 , find tan .y arc y

opposite

y

adjacent

Since y = arcsec x, then sec y = x

51

2


Derivatives: Inverse Trig Functions

sin y x

cos 1dy

ydx

1

cos

dy

dx y

2

1

1 sin

dy

dx y

2

1

1

dy

dx x

1sindy d

xdx dx

1siny x

1

2

1sin

1

dx

dx x

2 2sin cos 1x x

differentiate

solve for dy

dx

sinx y

differentiate substitute

derivative of inverse sine


Derivatives of Inverse Trig Functions

2

Let be a differentiable function of .

'arcsin

1

u x

d uu

dx u

2

'arccos

1

d uu

dx u

2

'arctan

1

d uu

dx u

2

'arccot

1

d uu

dx u

2

'arcsec

1

d uu

dx u u

2

'arccsc

1

d uu

dx u u


Model Problems

2

. arcsin 2

. arctan 3

. arcsin

. arcsec x

da x

dx

db x

dx

dc x

dx

dd e

dx

2

'arcsin

1

d uu

dx u

u = 2x

2 2

2 2

1 41 2 xx

2

'arctan

1

d uu

dx u

u = 3x

2 2

3 3

1 91 3 xx

1 2

2 2

1 / 2 1

21

x

x xx

2

'arcsin

1

d uu

dx u

u = x1/2

2

'arcsec

1

d uu

dx u u

u = e2x

2

22 2

2

1

x

x x

e

e e

4

2

1xe


Model Problems

1 2

1

1 2

. sin

. tan 5

. sec

da x

dx

db x

dx

dc x x

dx

2

'arcsin

1

d uu

dx u

u = x2

2 42

2 2

11

x x

xx

2

'arctan

1

d uu

dx u

u = 5x

2 2

5 5

1 251 5 xx

2 2

2 4 3 2

2 1

1

2 1

2 1

x

x x x x

x

x x x x x

u = x2 - x

2

'arcsec

1

d uu

dx u u


Elementary Functions

Elementary functions have proven sufficient to model most phenomena in

physics, chemistry, biology, engineering, economics, etc. An elementary function is a function from the following list or one that can be formed as a sum, product, quotient

or composition of functions in the list.

Algebraic

Polynomial

Rational

Functions w/radicals

TranscendentalLogarithmic

Exponential

Trigonometric

Inverse Trigonometric



0º 30º 45º 60º 90º

sin 0

cos 1

tan 0

1

2

1

2

1

2

2

2

2

3

2

3

2

3

3

3

1

0

UND.

aim: differentiate inverse trig functions course: calculus do now: aim: how do we differentiate...

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