Inverse Trig. Functions & Differentiation Section 5.8.

Inverse Trig. Functions &

Differentiation

Section 5.8

•Here, you can see that the sine function

y = sin x is not one-to-one.

–Use the Horizontal Line Test.

INVERSE TRIGONOMETRIC FUNCTIONS

•However, here, you can see that

the function f(x) = sin x, ,

is one-to-one.2 2x

INVERSE TRIGONOMETRIC FUNCTIONS

• As the definition of an inverse function states

• that

• we have:

•

• Thus, if -1 ≤ x ≤ 1, sin-1x is the number between and whose sine is x.

1 ( ) ( )f x y f y x

1sin sin and2 2

x y y x y

2 2

INVERSE SINE FUNCTIONS Equation 1

• Evaluate:

• a.

• b.

INVERSE SINE FUNCTIONS Example 1

1 1sin

2

1tan(arcsin )

3

Solve.1

1. arcsin2

y 32. arccos

2

33. sin arctan

4

Solve.1

1. arcsin2

y 32. arccos

2

33. sin arctan

4

1sin

2y

6ref

sin QI

6y

3cos

2x

x

6ref

cos QII

5

6x

3arctan

4

x

sin x

3tan

4

3

4

5

oppx

hyp

3

5x

tan QI

QI

• We have:

–This is because , and lies between and .

1 1sin

2 6

sin / 6 1/ 2 / 2 / 2

Example 1 aINVERSE SINE FUNCTIONS

/ 6

• Let , so .

–Then, we can draw a right triangle with angle θ.–So, we deduce from the Pythagorean Theorem

that the third side has length .

1arcsin

3

1sin

3

9 1 2 2

Example 1 bINVERSE SINE FUNCTIONS

–This enables us to read from the triangle that:

INVERSE SINE FUNCTIONS

1 1tan(arcsin ) tan

3 2 2

Example 1b

• In this case, the cancellation equations

for inverse functions become:

1

1

sin (sin ) for2 2

sin(sin ) for 1 1

x x x

x x x

INVERSE SINE FUNCTIONS Equations 2

• The graph is obtained from that of • the restricted sine function by reflection • about the line y = x.

INVERSE SINE FUNCTIONS

•We know that:

–The sine function f is continuous, so the inverse sine function is also continuous.

–The sine function is differentiable, so the inverse sine function is also differentiable (from Section 3.4).

INVERSE SINE FUNCTIONS

•since we know that is sin-1 differentiable, we

can just as easily calculate it by implicit

differentiation as follows.

INVERSE SINE FUNCTIONS

•Let y = sin-1x.

–Then, sin y = x and –π/2 ≤ y ≤ π/2.

–Differentiating sin y = x implicitly with respect to x,we obtain:

INVERSE SINE FUNCTIONS

cos 1

1

cos

dyy

dxdy

dx y

and

• Now, cos y ≥ 0 since –π/2 ≤ y ≤ π/2, so

INVERSE SINE FUNCTIONS

2 2

2

cos 1 sin 1

1 1

cos 1

y y x

dy

dx y x

Therefore

1

2

1(sin ) 1 1

1

dx x

dx x

Formula 3

• If f(x) = sin-1(x2 – 1), find:

(a) the domain of f.

(b) f ’(x).

INVERSE SINE FUNCTIONS Example 2

•Since the domain of the inverse sine

function is [-1, 1], the domain of f is:

INVERSE SINE FUNCTIONS Example 2 a

2 2{ | 1 1 1} { 0 2| }

{ | | 2}

2

|

, 2

x x x x

x x

•Combining Formula 3 with the Chain Rule,

we have:

Example 2 bINVERSE SINE FUNCTIONS

2

2 2

4 2

2 4

1'( ) ( 1)

1 ( 1)

12

1 ( 2 1)

2

2

df x x

dxx

xx x

x

x x

•The inverse cosine function is handled

similarly. –The restricted cosine function f(x) = cos x, 0 ≤ x

≤ π, is one-to-one.

–So, it has an inverse function denoted by cos-1 or arccos.

1cos cos and 0x y y x y

INVERSE COSINE FUNCTIONS Equation 4

• The cancellation equations are:

INVERSE COSINE FUNCTIONS

1cos (cos ) for 0x x x

1cos(cos ) for 1 1x x x

Equation 5

•The inverse cosine function,cos-1,

has domain [-1, 1] and range ,

and is a continuous function.

[0, ]

INVERSE COSINE FUNCTIONS

•Its derivative is given by:

–The formula can be proved by the same method as for Formula 3.

INVERSE COSINE FUNCTIONS Formula 6

1

2

1(cos ) 1 1

1

dx x

dx x

•The inverse tangent function,

tan-1 = arctan, has domain and range

.( / 2, / 2)

INVERSE TANGENT FUNCTIONS

•We know that:

–So, the lines

are vertical asymptotes of the graph of tan.

/ 2x

INVERSE TANGENT FUNCTIONS

( / 2) ( / 2)lim tan lim tanand

x xx x

•The graph of tan-1 is obtained by reflecting

the graph of the restricted tangent function

about the line y = x.

–It follows that the lines y = π/2 and y = -π/2 are horizontal asymptotes of the graph of tan-1.

INVERSE TANGENT FUNCTIONS

Inverse Trig. Functions

None of the 6 basic trig. functions has an inverse unless you restrict their domains.

Function Domain Range

y = arcsin x -1< x < 1 I & IV

y = arccos x -1< x < 1 I & II

y = arctan x < x < I & IV

y= arccot x < x < I & I

y = arcsec x I & II

y = arccsc I & IV

1x

1x

The Inverse Trigonometric FunctionsGraphs of six inverse trigonometric functions :

The Inverse Trigonometric FunctionsGraphs of six inverse trigonometric functions :

Inverse Properties

f (f –1(x)) = x and f –1(f (x)) = x

Remember that the trig. functions have inverses only in restricted domains.

•

Table 11DERIVATIVES

1 1

2 2

1 1

2 2

1 12 2

1 1(sin ) (csc )

1 1

1 1(cos ) (sec )

1 1

1 1(tan ) (cot )

1 1

d dx x

dx dxx x x

d dx x

dx dxx x x

d dx x

dx x dx x

Derivatives of Inverse Trig. Functions

Let u be a differentiable function of x.

'

2arcsin

1

d uu

dx u

'

2arccos

1

d uu

dx u

2

'

1arctan

u

uu

dx

d

2

'

1cot

u

uuarc

dx

d

1

sec2

'

uu

uuarc

dx

d

1

csc2

'

uu

uuarc

dx

d

•Each of these formulas can be

combined with the Chain Rule.

•For instance, if u is a differentiable

function of x, then

DERIVATIVES

1

2

12

1(sin )

1and

1(tan )

1

d duu

dx dxu

d duu

dx u dx

• Differentiate:

DERIVATIVES Example 5

1

1a.

sin

b. ( ) arctan

yx

f x x x

•

DERIVATIVES Example 5 a

1 1

1 2 1

1 2 2

(sin )

(sin ) (sin )

1

(sin ) 1

dy dx

dx dxd

x xdx

x x

1

1

siny

x

•

DERIVATIVES Example 5 b

( ) arctanf x x x

1/ 2122

1'( ) ( ) arctan

1 ( )

arctan2(1 )

f x x x xx

xx

x

Find each derivative with respect to x.

2. arc sec 2f x x

3. arctany x x

Find each derivative with respect to x.

2. arc sec 2f x x

y

2x

1

24 1x

sec 2y xsec tan 2

dyy y

dx

2cos cotdy

y ydx

2

1 12

2 4 1x x

2

1

4 1

dy

dx x x

3. arctany x x y x

arctan x tan x

x

1

21 x

dy dx

dx dx

2sec 1d

dx

21 1d

xdx

2

1

1

d

dx x

2

1arctan

1x x

x

2arctan

1

xx

x

Find each derivative with respect to the given variable.

4. sin arccosh t t

5. 2arcsin 1f x x

Find each derivative with respect to the given variable.

4. sin arccosh t t

1

t

21 t

sinh t

5. 2arcsin 1f x x

1x 1

21 1x

' cosd

h tdt

cos t sin 1d

dt

21

11

t d

dt

2

1

1

d

dt t

2

1

1 1

t

t

21

t

t

2f x

' 2d

f xdx

sin 1x cos 1

d

dx

1

cos

d

dx

2

2

1 1x

2

1

1 1

d

dx x

Example 1

)2arcsin( ateDifferenti x

2)2(1

12x

241

2

x

)2arcsin( x

Example 2

2arctan ateDifferenti

x

2

2

2

4

2

412

1

21

21

1

2arctan

x

x

x

x

Example 2

2arctan ateDifferenti

x

2

2

2

4

2

412

1

21

21

1

2arctan

x

x

x

x

Example 2

2arctan ateDifferenti

x

2

2

2

4

2

412

1

21

21

1

2arctan

x

x

x

x

Example 2

2arctan ateDifferenti

x

2

2

2

4

2

412

1

21

21

1

2arctan

x

x

x

x

Example 2

xxarccos ateDifferenti

2

2

1arccos

1

1arccos

arccos

x

xx

xxx

xx

Example 2

xxarccos ateDifferenti

2

2

1arccos

1

1arccos

arccos

x

xx

xxx

xx

Example 2

xxarccos ateDifferenti

2

2

1arccos

1

1arccos

arccos

x

xx

xxx

xx

Some homework examples:

Write the expression in algebraic form

x3arctansec

Solution: Use the right triangle

xy

yx

3tan

3arctan

Now using the triangle we can find the hyp.

Letthen

y

3x

1

222 9131 xx 291 x

22

911

91sec x

xy

Some homework examples:

Find the derivative of:

2

1

arctanarctan)( xxxf

Let u = 2

1

x

xdx

du

2

1

xxx

xxf

12

1

1

2

1

)( 2

Example xx

dx

d 21sec xxu 2

12 xdx

du

dx

du

uu

1

12

1

12222

xxxx

x

Example x

dx

dsintan 1 xu sin

xdx

ducos

dx

du

u

21

1

x

x2sin1

cos

Find an equation for the line tangent to the graph of at

x = -1xy 1cot

21

1

x

xdx

d 1cot

2

1

)1(1

12

At x = -1

Slope of tangent line

When x = -1, y =4

3

12

1

4

3

xy