# 1 Inverse Trig Functions

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• Lecture 6Section 7.7 Inverse Trigonometric Functions Section

7.8 Hyperbolic Sine and Cosine

Jiwen He

1 Inverse Trig Functions

1.1 Inverse Sine

Inverse Since sin1 x (or arcsin x)

1

• domain:[ 12, 1

2] range:[1, 1]

2

• sin(sin1 x) = x

3

• 4

• domain:[1, 1] range:[ 12, 1

2]

Trigonometric Properties

5

• sin(sin1 x) = x cos(sin1 x) =

1 x2

tan(sin1 x) =x

1 x2cot(sin1 x) =

1 x2

x

sec(sin1 x) =1

1 x2csc(sin1 x) =

1x

Differentiation

Theorem 1.d

dxsin1 x =

11 x2

.

Proof.Let y = sin1 x. Then x = sin y,

d

dxsin1 x =

1ddy sin y

=1

cos y=

1cos(sin1 x)

=1

1 x2.

Theorem 2.

d

dxsin1 u =

11 u2

du

dx,

1

1 u2du = sin1 u + C

Integration: u-Substitution

6

• Theorem 3. g(x)

1 (g(x))2dx = sin1(g(x)) + C

ProofLet u = g(x). Then du = g(x) dx,

g(x)1 (g(x))2

dx =

11 u2

du = sin1 u + C = sin1(g(x)) + C

Examples 4.

14 x2

dx =

11 u2

du = sin1 u+C = sin1x

2+C. Note

that 4 x2 = 4(1

(x2

)2). Let u = x2 . Then du = 12dx. 12x x2 dx =1

1 u2du = sin1 u+C = sin1(x 1)+C. Note that 2xx2 = 1 (x2

2x + 1) = 1 (x 1)2 (complete the square). Let u = x 1. Then du = dx.

1.2 Inverse Tangent

Inverse Tangent tan1 x (or arctanx)

7

• y = tan x domain:( 12, 1

2) range:(,)

8

• Trigonometric Properties

tan(tan1 x) = x cot(tan1 x) =1x

sin(tan1 x) =x

1 + x2cos(tan1 x) =

11 + x2

sec(tan1 x) =

1 + x2 csc(tan1 x) =

1 + x2

x

Differentiation

Theorem 5.d

dxtan1 x =

11 + x2

.

Proof.Let y = tan1 x. Then x = tan y,

d

dxtan1 x =

1ddy tan y

=1

(sec y)2=

1(sec(tan1 x)

)2 = 11 + x2 .Theorem 6.

d

dxtan1 u =

11 + u2

du

dx,

1

1 + u2du = tan1 u + C

9

• Integration: u-Substitution

Theorem 7. g(x)

1 + (g(x))2dx = tan1(g(x)) + C

ProofLet u = g(x). Then du = g(x) dx,

g(x)1 + (g(x))2

dx =

11 + u2

du = tan1 u + C = tan1(g(x)) + C

Examples 8.

14 + x2

dx =12

1

1 + u2du =

12

tan1 u + C =12

tan1x

2+ C.

Note that 4+x2 = 4(1 +

(x2

)2). Let u = x2 . Then du = 12dx. 12 + 2x + x2 dx =1

1 + u2du = tan1(x+1)+C. Note that 2+2x+x2 = 1+(x2+2x+1) = 1+(x+

1)2 (complete the square). Let u = x + 1. Then du = dx.

ex

1 + e2xdx =

11 + u2

du = tan1(ex) + C. Note that 1 + e2x = 1 + (ex)2 (complete

the square). Let u = ex. Then du = exdx.

Quiz

Quiz

10

• Let f (t) = kf(t).

1. For f(0) = 4, f(t) =: (a) kt + 4, (b) 4ekt, (c) 4ekt.

2. For k > 0, double time T =: (a)4k

, (b)ln 2k

(c) ln 2k

.

1.3 Inverse Secant

Inverse Secant sec1 x

11

• y = sec x domain:[0, 12)( 1

2, ] range:(,1][1,)

12

• Trigonometric Properties

sec(sec1 x) = x csc(sec1 x) =x

x2 1

sin(sec1 x) =

x2 1x

cos(sec1 x) =1x

tan(sec1 x) =

x2 1 cot(sec1 x) = 1x2 1

Differentiation

Theorem 9.d

dxsec1 x =

1|x|

x2 1.

Proof.Let y = sec1 x. Then x = sec y,

d

dxsec1 x =

1ddy sec y

=1

(sec y tan y)2=

1|x|

x2 1.

Theorem 10.

d

dxsec1 u =

1|u|

u2 1du

dx,

1

u

u2 1du = sec1 |u|+ C

13

• Integration: u-Substitution

Theorem 11. g(x)

g(x)

(g(x))2 1dx = sec1(|g(x)|) + C

ProofLet u = g(x). Then du = g(x) dx,

g(x)g(x)

(g(x))2 1

dx =

1u

u2 1du = sec1(|g(x)|) + C

Examples 12.

1x

x 1dx = 2

1

u

u2 1du =

12

sec1

x + C. Note that

x 1 = (

x)2 1. Let u =

x. Then x = u2, dx = 2udu.

1.4 Other Trig Inverses

Other Trigonometric Inverses

Other Trigonometric Inverses

14

• sin1 x + cos1 x =

2or cos1 x =

2 sin1 x

tan1 x + cot1 x =

2or cot1 x =

2 tan1 x

sec1 x + csc1 x =

2or csc1 x =

2 sec1 x

Differentiation

Theorem 13.

d

dxcos1 x = d

dxsin1 x = 1

1 x2d

dxcot1 x = d

dxtan1 x = 1

1 + x2d

dxcsc1 x = d

dxsec1 x = 1

|x|

x2 1

Quiz (cont.)The value, at the end of the 4 years, of a principle of \$100 invested at 4%

compounded

3. annually: (a) 400(1 + 0.04), (b) 100(1 + 0.04)4, (c) 100(1 + 0.16).

4. continuously: (a) 100e0.04, (b) 100e0.16, (c) 100(1 + 0.04)4.

2 Hyperbolic Sine and Cosine

2.1 Definition

Hyperbolic Sine and Cosine

15

• Definition 14.

sinhx =12(ex ex

), coshx =

12(ex + ex

)Theorem 15.

d

dxsinhx = cosh,

d

dxcoshx = sinh,

Identities

16

• 17

• cosh2 x sinh2 x = 1sinh(x + y) = sinhx cosh y + coshx sinh ycosh(x + y) = coshx cosh y + sinhx sinh y

cos2 x + sin2 x = 1sin(x + y) = sin x cos y + cos x sin ycos(x + y) = cos x cos y sinx sin y

Outline

Contents

1 Inverse Trig Functions 1

18

• 1.1 Inverse Sine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Inverse Tangent . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Inverse Secant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Other Trig Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Hyperbolic Sine and Cosine 152.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

19