lesson 16: inverse trigonometric functions (section 041 slides)

Section 3.5Inverse Trigonometric

FunctionsV63.0121.041, Calculus I

New York University

November 1, 2010

Announcements

I Midterm grades have been submittedI Quiz 3 this week in recitation on Section 2.6, 2.8, 3.1, 3.2I Thank you for the evaluations

. . . . . .

. . . . . .

Announcements

I Midterm grades have beensubmitted

I Quiz 3 this week inrecitation on Section 2.6,2.8, 3.1, 3.2

I Thank you for theevaluations

V63.0121.041, Calculus I (NYU) Section 3.5 Inverse Trigonometric Functions November 1, 2010 2 / 32

. . . . . .

Objectives

I Know the definitions,domains, ranges, andother properties of theinverse trignometricfunctions: arcsin, arccos,arctan, arcsec, arccsc,arccot.

I Know the derivatives of theinverse trignometricfunctions.


. . . . . .

What is an inverse function?

DefinitionLet f be a function with domain D and range E. The inverse of f is thefunction f−1 defined by:

f−1(b) = a,

where a is chosen so that f(a) = b.

Sof−1(f(x)) = x, f(f−1(x)) = x


. . . . . .

What functions are invertible?

In order for f−1 to be a function, there must be only one a in Dcorresponding to each b in E.

I Such a function is called one-to-oneI The graph of such a function passes the horizontal line test: any

horizontal line intersects the graph in exactly one point if at all.I If f is continuous, then f−1 is continuous.


. . . . . .

Outline

Inverse Trigonometric Functions

Derivatives of Inverse Trigonometric FunctionsArcsineArccosineArctangentArcsecant

Applications


. . . . . .

arcsin

Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].

. .x

.y

.sin.

.−π

2

.

.π

2

.y = x

.

. .arcsin

I The domain of arcsin is [−1,1]I The range of arcsin is

[−π

2,π

2

]


. . . . . .

arcsin

Arcsin is the inverse of the sine function after restriction to [−π/2, π/2].

. .x

.y

.sin.

.

.

.−π

2

.

.π

2

.y = x

.

. .arcsin

I The domain of arcsin is [−1,1]I The range of arcsin is

[−π

2,π

2

]


. . . . . .

arccos

Arccos is the inverse of the cosine function after restriction to [0, π]

. .x

.y

.cos..0

..π

.y = x

.

. .arccos

I The domain of arccos is [−1,1]I The range of arccos is [0, π]


. . . . . .

arccos

Arccos is the inverse of the cosine function after restriction to [0, π]

. .x

.y

.cos.

.

..0

..π

.y = x

.

. .arccos

I The domain of arccos is [−1,1]I The range of arccos is [0, π]


. . . . . .

arctan

Arctan is the inverse of the tangent function after restriction to[−π/2, π/2].

. .x

.y

.tan

.−3π2

.−π

2.π

2 .3π2

.y = x

.arctan

.−π

2

.π

2

I The domain of arctan is (−∞,∞)

I The range of arctan is(−π

2,π

2

)I lim

x→∞arctan x =

π

2, limx→−∞

arctan x = −π

2


. . . . . .

arcsec

Arcsecant is the inverse of secant after restriction to[0, π/2) ∪ (π,3π/2].

. .x

.y

.sec

.−3π2

.−π

2.π

2 .3π2

.y = x

.

.

.π

2

.3π2

I The domain of arcsec is (−∞,−1] ∪ [1,∞)

I The range of arcsec is[0,

π

2

)∪(π2, π]

I limx→∞

arcsec x =π

2, limx→−∞

arcsec x =3π2


. . . . . .

arcsec

Arcsecant is the inverse of secant after restriction to[0, π/2) ∪ (π,3π/2].

. .x

.y

.sec

.−3π2

.−π

2.π

2 .3π2

.

.

.y = x

.

.

.π

2

.3π2

I The domain of arcsec is (−∞,−1] ∪ [1,∞)

I The range of arcsec is[0,

π

2

)∪(π2, π]

I limx→∞

arcsec x =π

2, limx→−∞

arcsec x =3π2


. . . . . .

Values of Trigonometric Functions

x 0π

6π

4π

3π

2

sin x 0 12

√22

√32

1

cos x 1√32

√22

12

0

tan x 01√3

1√3 undef

cot x undef√3 1

1√3

0

sec x 12√3

2√2

2 undef

csc x undef 22√2

2√3

1


. . . . . .

Check: Values of inverse trigonometric functions

Example

FindI arcsin(1/2)I arctan(−1)

I arccos

(−√22

)

Solution

Iπ

6I −π

4I

3π4


. . . . . .


Example


I arccos

(−√22

)

Solution

Iπ

6

I −π

4I

3π4


. . . . . .

What is arctan(−1)?

. .

.

..3π/4

..−π/4

.sin(3π/4) =√22

.cos(3π/4) = −√22.sin(π/4) = −

√22

.cos(π/4) =√22

I Yes, tan(3π4

)= −1

I But, the range of arctan is(−π

2,π

2

)I Another angle whose

tangent is −1 is −π

4, and

this is in the right range.I So arctan(−1) = −π

4


. . . . . .


. .

.

..3π/4

..−π/4

.sin(3π/4) =√22

.cos(3π/4) = −√22

.sin(π/4) = −√22

.cos(π/4) =√22

I Yes, tan(3π4

)= −1


2,π

2



4, and


4


. . . . . .


. .

.

..3π/4

..−π/4

.sin(3π/4) =√22

.cos(3π/4) = −√22

.sin(π/4) = −√22

.cos(π/4) =√22

I Yes, tan(3π4

)= −1


2,π

2

)

I Another angle whosetangent is −1 is −π

4, and


4


. . . . . .


. .

.

..3π/4

..−π/4

.sin(3π/4) =√22

.cos(3π/4) = −√22

.sin(π/4) = −√22

.cos(π/4) =√22

I Yes, tan(3π4

)= −1


2,π

2



4, and

this is in the right range.

I So arctan(−1) = −π

4


. . . . . .


. .

.

..3π/4

..−π/4

.sin(3π/4) =√22

.cos(3π/4) = −√22

.sin(π/4) = −√22

.cos(π/4) =√22

I Yes, tan(3π4

)= −1


2,π

2



4, and


4


. . . . . .


Example


I arccos

(−√22

)

Solution

Iπ

6I −π

4

I3π4


. . . . . .


Example


I arccos

(−√22

)

Solution

Iπ

6I −π

4I

3π4


. . . . . .

Caution: Notational ambiguity

..sin2 x = (sin x)2 .sin−1 x = (sin x)−1

I sinn x means the nth power of sin x, except when n = −1!I The book uses sin−1 x for the inverse of sin x, and never for

(sin x)−1.

I I use csc x for1

sin xand arcsin x for the inverse of sin x.


. . . . . .

Outline



Applications


. . . . . .

The Inverse Function Theorem

Theorem (The Inverse Function Theorem)

Let f be differentiable at a, and f′(a) ̸= 0. Then f−1 is defined in anopen interval containing b = f(a), and

(f−1)′(b) =1

f′(f−1(b))

In Leibniz notation we have

dxdy

=1

dy/dx

Upshot: Many times the derivative of f−1(x) can be found by implicitdifferentiation and the derivative of f:


. . . . . .

Illustrating the Inverse Function Theorem.

.

Example

Use the inverse function theorem to find the derivative of the square rootfunction.

Solution (Newtonian notation)

Let f(x) = x2 so that f−1(y) =√y. Then f′(u) = 2u so for any b > 0 we have

(f−1)′(b) =1

2√b

Solution (Leibniz notation)

If the original function is y = x2, then the inverse function is defined by x = y2.Differentiate implicitly:

1 = 2ydydx

=⇒ dydx

=1

2√x


. . . . . .

Derivation: The derivative of arcsin

Let y = arcsin x, so x = sin y. Then

cos ydydx

= 1 =⇒ dydx

=1

cos y=

1cos(arcsin x)

To simplify, look at a righttriangle:

cos(arcsin x) =√1− x2

So

ddx

arcsin(x) =1√

1− x2 .

.1 .x

. .y = arcsin x

.√1− x2


. . . . . .



cos ydydx

= 1 =⇒ dydx

=1

cos y=

1cos(arcsin x)



So

ddx

arcsin(x) =1√

1− x2

.

.1 .x

. .y = arcsin x

.√1− x2


. . . . . .



cos ydydx

= 1 =⇒ dydx

=1

cos y=

1cos(arcsin x)



So

ddx

arcsin(x) =1√

1− x2 .

.1 .x

. .y = arcsin x

.√1− x2


. . . . . .

Graphing arcsin and its derivative

I The domain of f is [−1,1],but the domain of f′ is(−1,1)

I limx→1−

f′(x) = +∞

I limx→−1+

f′(x) = +∞ ..|.−1

.|.1

.

. .arcsin

.1√

1− x2


. . . . . .

Composing with arcsin

Example

Let f(x) = arcsin(x3 + 1). Find f′(x).

SolutionWe have

ddx

arcsin(x3 + 1) =1√

1− (x3 + 1)2ddx

(x3 + 1)

=3x2√

−x6 − 2x3


. . . . . .

Derivation: The derivative of arccos

Let y = arccos x, so x = cos y. Then

− sin ydydx

= 1 =⇒ dydx

=1

− sin y=

1− sin(arccos x)


sin(arccos x) =√1− x2

So

ddx

arccos(x) = − 1√1− x2 .

.1.√1− x2

.x. .y = arccos x


. . . . . .

Graphing arcsin and arccos

..|.−1

.|.1

.

. .arcsin

.

. .arccos

Note

cos θ = sin(π2− θ)

=⇒ arccos x =π

2− arcsin x

So it’s not a surprise that theirderivatives are opposites.


. . . . . .

Graphing arcsin and arccos

..|.−1

.|.1

.

. .arcsin

.

. .arccosNote

cos θ = sin(π2− θ)

=⇒ arccos x =π

2− arcsin x

So it’s not a surprise that theirderivatives are opposites.


. . . . . .

Derivation: The derivative of arctan

Let y = arctan x, so x = tan y. Then

sec2 ydydx

= 1 =⇒ dydx

=1

sec2 y= cos2(arctan x)


cos(arctan x) =1√

1+ x2

So

ddx

arctan(x) =1

1+ x2 .

.x

.1. .y = arctan x

.√1+ x2


. . . . . .



sec2 ydydx

= 1 =⇒ dydx

=1



cos(arctan x) =1√

1+ x2

So

ddx

arctan(x) =1

1+ x2

.

.x

.1. .y = arctan x

.√1+ x2


. . . . . .



sec2 ydydx

= 1 =⇒ dydx

=1



cos(arctan x) =1√

1+ x2

So

ddx

arctan(x) =1

1+ x2

.

.x

.1

. .y = arctan x

.√1+ x2


. . . . . .



sec2 ydydx

= 1 =⇒ dydx

=1



cos(arctan x) =1√

1+ x2

So

ddx

arctan(x) =1

1+ x2

.

.x

.1. .y = arctan x

.√1+ x2


. . . . . .



sec2 ydydx

= 1 =⇒ dydx

=1



cos(arctan x) =1√

1+ x2

So

ddx

arctan(x) =1

1+ x2 .

.x

.1. .y = arctan x

.√1+ x2


. . . . . .

Graphing arctan and its derivative

. .x

.y

.arctan

.1

1+ x2

.π/2

.−π/2

I The domain of f and f′ are both (−∞,∞)

I Because of the horizontal asymptotes, limx→±∞

f′(x) = 0


. . . . . .

Composing with arctan

Example

Let f(x) = arctan√x. Find f′(x).

Solution

ddx

arctan√x =

11+

(√x)2 d

dx√x =

11+ x

· 12√x

=1

2√x+ 2x

√x


. . . . . .

Derivation: The derivative of arcsec

Try this first.

Let y = arcsec x, so x = sec y. Then

sec y tan ydydx

= 1 =⇒ dydx

=1

sec y tan y=

1x tan(arcsec(x))


tan(arcsec x) =√x2 − 11

So

ddx

arcsec(x) =1

x√x2 − 1 .

.x

.1. .y = arcsec x

.√

x2 − 1


. . . . . .


Try this first. Let y = arcsec x, so x = sec y. Then

sec y tan ydydx

= 1 =⇒ dydx

=1

sec y tan y=

1x tan(arcsec(x))



So

ddx

arcsec(x) =1

x√x2 − 1 .

.x

.1. .y = arcsec x

.√

x2 − 1


. . . . . .



sec y tan ydydx

= 1 =⇒ dydx

=1

sec y tan y=

1x tan(arcsec(x))



So

ddx

arcsec(x) =1

x√x2 − 1

.

.x

.1. .y = arcsec x

.√

x2 − 1


. . . . . .



sec y tan ydydx

= 1 =⇒ dydx

=1

sec y tan y=

1x tan(arcsec(x))



So

ddx

arcsec(x) =1

x√x2 − 1

.

.x

.1

. .y = arcsec x

.√

x2 − 1


. . . . . .



sec y tan ydydx

= 1 =⇒ dydx

=1

sec y tan y=

1x tan(arcsec(x))



So

ddx

arcsec(x) =1

x√x2 − 1

.

.x

.1. .y = arcsec x

.√

x2 − 1


. . . . . .



sec y tan ydydx

= 1 =⇒ dydx

=1

sec y tan y=

1x tan(arcsec(x))



So

ddx

arcsec(x) =1

x√x2 − 1 .

.x

.1. .y = arcsec x

.√

x2 − 1


. . . . . .

Another Example

Example

Let f(x) = earcsec 3x. Find f′(x).

Solution

f′(x) = earcsec 3x · 13x√

(3x)2 − 1· 3

=3earcsec 3x

3x√9x2 − 1


. . . . . .

Outline



Applications


. . . . . .

Application

ExampleOne of the guiding principles ofmost sports is to “keep youreye on the ball.” In baseball, abatter stands 2 ft away fromhome plate as a pitch is thrownwith a velocity of 130 ft/sec(about 90mph). At what ratedoes the batter’s angle of gazeneed to change to follow theball as it crosses home plate?

SolutionLet y(t) be the distance from the ball to home plate, and θ the angle thebatter’s eyes make with home plate while following the ball. We knowy′ = −130 and we want θ′ at the moment that y = 0.


. . . . . .

SolutionLet y(t) be the distance from the ball to home plate, and θ the angle thebatter’s eyes make with home plate while following the ball. We knowy′ = −130 and we want θ′ at the moment that y = 0.

We have θ = arctan(y/2). Thus

dθdt

=1

1+ (y/2)2· 12dydt

When y = 0 and y′ = −130,then

dθdt

∣∣∣∣y=0

=1

1+ 0·12(−130) = −65 rad/sec

The human eye can only trackat 3 rad/sec!

..2 ft

.y

.130 ft/sec

.

.θ


. . . . . .

SolutionLet y(t) be the distance from the ball to home plate, and θ the angle thebatter’s eyes make with home plate while following the ball. We knowy′ = −130 and we want θ′ at the moment that y = 0.We have θ = arctan(y/2). Thus

dθdt

=1

1+ (y/2)2· 12dydt

When y = 0 and y′ = −130,then

dθdt

∣∣∣∣y=0

=1

1+ 0·12(−130) = −65 rad/sec

The human eye can only trackat 3 rad/sec!

..2 ft

.y

.130 ft/sec

.

.θ


. . . . . .

Summary

y y′

arcsin x1√

1− x2

arccos x − 1√1− x2

arctan x1

1+ x2

arccot x − 11+ x2

arcsec x1

x√x2 − 1

arccsc x − 1x√x2 − 1

I Remarkable that thederivatives of thesetranscendental functionsare algebraic (or evenrational!)


lesson 16: inverse trigonometric functions (section 041 slides)

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