# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 8

## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 8

Published $\text {1968}$.

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## $8 \quad$ Hyperbolic Functions

### Definition of Hyperbolic Functions

$8.1$: Hyperbolic Sine
$8.2$: Hyperbolic Cosine
$8.3$: Hyperbolic Tangent
$8.4$: Hyperbolic Cotangent
$8.5$: Hyperbolic Secant
$8.6$: Hyperbolic Cosecant

### Relationships among Hyperbolic Functions

$8.7$: Hyperbolic Tangent: $\tanh x := \dfrac {\sinh x} {\cosh x}$
$8.8$: Hyperbolic Cotangent is Reciprocal of Hyperbolic Tangent: $\coth x = \dfrac 1 {\tanh x}$
$8.8$: Hyperbolic Cotangent: $\coth x := \dfrac {\cosh x} {\sinh x}$
$8.9$: Hyperbolic Secant: $\sech x := \dfrac 1 {\cosh x}$
$8.10$: Hyperbolic Cosecant: $\csch x := \dfrac 1 {\sinh x}$
$8.11$: Difference of Squares of Hyperbolic Cosine and Sine
$8.12$: Sum of Squares of Hyperbolic Secant and Tangent
$8.13$: Difference of Squares of Hyperbolic Cotangent and Cosecant

### Functions of Negative Arguments

$8.14$: Hyperbolic Sine Function is Odd
$8.15$: Hyperbolic Cosine Function is Even
$8.16$: Hyperbolic Tangent Function is Odd
$8.17$: Hyperbolic Cosecant Function is Odd
$8.18$: Hyperbolic Secant Function is Even
$8.19$: Hyperbolic Cotangent Function is Odd

$8.20$: Hyperbolic Sine of Sum
Hyperbolic Sine of Difference
$8.21$: Hyperbolic Cosine of Sum
Hyperbolic Cosine of Difference
$8.22$: Hyperbolic Tangent of Sum
Hyperbolic Tangent of Difference
$8.23$: Hyperbolic Cotangent of Sum
Hyperbolic Cotangent of Difference

### Double Angle Formulas

$8.24$: Double Angle Formula for Hyperbolic Sine
$8.25$: Double Angle Formula for Hyperbolic Cosine
Double Angle Formula for Hyperbolic Cosine: Corollary 1
Double Angle Formula for Hyperbolic Cosine: Corollary 2
$8.26$: Double Angle Formula for Hyperbolic Tangent

### Half Angle Formulas

$8.27$: Half Angle Formula for Hyperbolic Sine
$8.28$: Half Angle Formula for Hyperbolic Cosine
$8.29$: Half Angle Formula for Hyperbolic Tangent
Half Angle Formula for Hyperbolic Tangent: Corollary 1
Half Angle Formula for Hyperbolic Tangent: Corollary 2

### Multiple Angle Formulas

$8.30$: Triple Angle Formula for Hyperbolic Sine
$8.31$: Triple Angle Formula for Hyperbolic Cosine
$8.32$: Triple Angle Formula for Hyperbolic Tangent
$8.33$: Quadruple Angle Formula for Hyperbolic Sine
$8.34$: Quadruple Angle Formula for Hyperbolic Cosine
$8.35$: Quadruple Angle Formula for Hyperbolic Tangent

### Powers of Hyperbolic Functions

$8.36$: Square of Hyperbolic Sine
$8.37$: Square of Hyperbolic Cosine
$8.38$: Cube of Hyperbolic Sine
$8.39$: Cube of Hyperbolic Cosine
$8.40$: Fourth Power of Hyperbolic Sine
$8.41$: Fourth Power of Hyperbolic Cosine

### Sum, Difference and Product of Hyperbolic Functions

$8.42$: Prosthaphaeresis Formula for Hyperbolic Sine plus Hyperbolic Sine
$8.43$: Prosthaphaeresis Formula for Hyperbolic Sine minus Hyperbolic Sine
$8.44$: Prosthaphaeresis Formula for Hyperbolic Cosine plus Hyperbolic Cosine
$8.45$: Prosthaphaeresis Formula for Hyperbolic Cosine minus Hyperbolic Cosine
$8.46$: Simpson's Formula for Hyperbolic Sine by Hyperbolic Sine
$8.47$: Simpson's Formula for Hyperbolic Cosine by Hyperbolic Cosine
$8.48$: Simpson's Formula for Hyperbolic Sine by Hyperbolic Cosine

Not documented.

### Graphs of Hyperbolic Functions

$8.49$: Graph of Hyperbolic Sine Function
$8.50$: Graph of Hyperbolic Cosine Function
$8.51$: Graph of Hyperbolic Tangent Function
$8.52$: Graph of Hyperbolic Cotangent Function
$8.53$: Graph of Hyperbolic Secant Function
$8.54$: Graph of Hyperbolic Cosecant Function

### Inverse Hyperbolic Functions

$8.55$: Inverse Hyperbolic Sine
$8.56$: Inverse Hyperbolic Cosine
$8.57$: Inverse Hyperbolic Tangent
$8.59$: Inverse Hyperbolic Secant
$8.60$: Inverse Hyperbolic Cosecant

### Relations Between Inverse Hyperbolic Functions

$8.61$: Real Area Hyperbolic Sine of Reciprocal equals Real Area Hyperbolic Cosecant
$8.62$: Real Area Hyperbolic Cosine of Reciprocal equals Real Area Hyperbolic Secant
$8.63$: Real Area Hyperbolic Tangent of Reciprocal equals Real Area Hyperbolic Cotangent
$8.64$: Inverse Hyperbolic Sine is Odd Function
$8.65$: Inverse Hyperbolic Tangent is Odd Function
$8.66$: Inverse Hyperbolic Cotangent is Odd Function
$8.67$: Inverse Hyperbolic Cosecant is Odd Function

### Graphs of Inverse Hyperbolic Functions

$8.68$: Graph of Inverse Hyperbolic Sine
$8.69$: Graph of Inverse Hyperbolic Cosine
$8.70$: Graph of Inverse Hyperbolic Tangent
$8.71$: Graph of Inverse Hyperbolic Cotangent
$8.72$: Graph of Inverse Hyperbolic Secant
$8.73$: Graph of Inverse Hyperbolic Cosecant

### Relations Between Hyperbolic and Trigonometric Functions

$8.74$: Hyperbolic Sine in terms of Sine
$8.75$: Hyperbolic Cosine in terms of Cosine
$8.76$: Hyperbolic Tangent in terms of Tangent
$8.77$: Hyperbolic Cosecant in terms of Cosecant
$8.78$: Hyperbolic Secant in terms of Secant
$8.79$: Hyperbolic Cotangent in terms of Cotangent
$8.80$: Sine in terms of Hyperbolic Sine
$8.81$: Cosine in terms of Hyperbolic Cosine
$8.82$: Tangent in terms of Hyperbolic Tangent
$8.83$: Cosecant in terms of Hyperbolic Cosecant
$8.84$: Secant in terms of Hyperbolic Secant
$8.85$: Cotangent in terms of Hyperbolic Cotangent

### Periodicity of Hyperbolic Functions

$8.86$: Periodicity of Hyperbolic Sine
$8.87$: Periodicity of Hyperbolic Cosine
$8.88$: Periodicity of Hyperbolic Tangent
$8.89$: Periodicity of Hyperbolic Cosecant
$8.90$: Periodicity of Hyperbolic Secant
$8.91$: Periodicity of Hyperbolic Cotangent

### Relations Between Inverse Hyperbolic and Inverse Trigonometric Functions

$8.92$: Inverse Sine of Imaginary Number
$8.93$: Inverse Hyperbolic Sine of Imaginary Number
$8.94$: Inverse Cosine of Imaginary Number
$8.95$: Inverse Hyperbolic Cosine of Imaginary Number
$8.96$: Inverse Tangent of Imaginary Number
$8.97$: Inverse Hyperbolic Tangent of Imaginary Number
$8.98$: Inverse Cotangent of Imaginary Number
$8.99$: Inverse Hyperbolic Cotangent of Imaginary Number
$8.100$: Inverse Secant of Imaginary Number
$8.101$: Inverse Hyperbolic Secant of Imaginary Number
$8.102$: Inverse Cosecant of Imaginary Number
$8.103$: Inverse Hyperbolic Cosecant of Imaginary Number

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