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.


•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).


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

can just as easily calculate it by implicit

differentiation as follows.


•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:


cos 1

1

cos

dyy

dxdy

dx y

and

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


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


( / 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 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 :

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

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

Inverse Trig. Functions & Differentiation Section 5.8.

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