# Book:Murray R. Spiegel/Mathematical Handbook of Formulas and Tables/Chapter 5

## Murray R. Spiegel: Mathematical Handbook of Formulas and Tables: Chapter 5

Published $\text {1968}$.

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## $5 \quad$ Trigonometric Functions

### Definition of Trigonometric Functions for a Right Triangle

$5.1$ Sine of Angle
$5.2$ Cosine of Angle
$5.3$ Tangent of Angle
$5.4$ Cotangent of Angle
$5.5$ Secant of Angle
$5.6$ Cosecant of Angle

### Extensions to Angles which may be Greater than $90 \degrees$

Cartesian Coordinate Plane
Cartesian Coordinate Pair:
$x$ Coordinate
$y$ Coordinate
Distance of Point from Origin in Cartesian Coordinates
Polar Coordinates:
Angular Coordinate
Quadrant $\text I$
Quadrant $\text {II}$
Quadrant $\text {III}$
Quadrant $\text {IV}$

$5.7$ Sine of Angle in Cartesian Plane
$5.8$ Cosine of Angle in Cartesian Plane
$5.9$ Tangent of Angle in Cartesian Plane
$5.10$ Cotangent of Angle in Cartesian Plane
$5.11$ Secant of Angle in Cartesian Plane
$5.12$ Cosecant of Angle in Cartesian Plane

### Relationship between Degrees and Radians

$5.13$ Value of Radian in Degrees
$5.14$ Value of Degree in Radians

### Relationships among Trigonometric Functions

$5.15$ Tangent is Sine divided by Cosine
$5.16$ Cotangent is Reciprocal of Tangent
$5.16$ Cotangent is Cosine divided by Sine
$5.17$ Secant is Reciprocal of Cosine
$5.18$ Cosecant is Reciprocal of Sine
$5.19$ Sum of Squares of Sine and Cosine
$5.20$ Difference of Squares of Secant and Tangent
$5.21$ Difference of Squares of Cosecant and Cotangent

### Signs and Variations of Trigonometric Functions

Shape of Sine Function
Shape of Cosine Function
Shape of Tangent Function
Shape of Cotangent Function
Shape of Secant Function
Shape of Cosecant Function

### Exact Values for Trigonometric Functions of Various Angles

Angle in Degrees Angle in Radians Particular Values of Sine Function Particular Values of Cosine Function Particular Values of Tangent Function Particular Values of Cotangent Function Particular Values of Secant Function Particular Values of Cosecant Function
$0 \degrees$ $0$ Sine of $0 \degrees$ Cosine of $0 \degrees$ Tangent of $0 \degrees$ Cotangent of $0 \degrees$ Secant of $0 \degrees$ Cosecant of $0 \degrees$
$15 \degrees$ $\dfrac \pi {12}$ Sine of $15 \degrees$ Cosine of $15 \degrees$ Tangent of $15 \degrees$ Cotangent of $15 \degrees$ Secant of $15 \degrees$ Cosecant of $15 \degrees$
$30 \degrees$ $\dfrac \pi 6$ Sine of $30 \degrees$ Cosine of $30 \degrees$ Tangent of $30 \degrees$ Cotangent of $30 \degrees$ Secant of $30 \degrees$ Cosecant of $30 \degrees$
$45 \degrees$ $\dfrac \pi 4$ Sine of $45 \degrees$ Cosine of $45 \degrees$ Tangent of $45 \degrees$ Cotangent of $45 \degrees$ Secant of $45 \degrees$ Cosecant of $45 \degrees$
$60 \degrees$ $\dfrac \pi 3$ Sine of $60 \degrees$ Cosine of $60 \degrees$ Tangent of $60 \degrees$ Cotangent of $60 \degrees$ Secant of $60 \degrees$ Cosecant of $60 \degrees$
$75 \degrees$ $\dfrac {5 \pi} {12}$ Sine of $75 \degrees$ Cosine of $75 \degrees$ Tangent of $75 \degrees$ Cotangent of $75 \degrees$ Secant of $75 \degrees$ Cosecant of $75 \degrees$
$90 \degrees$ $\dfrac \pi 2$ Sine of $90 \degrees$ Cosine of $90 \degrees$ Tangent of $90 \degrees$ Cotangent of $90 \degrees$ Secant of $90 \degrees$ Cosecant of $90 \degrees$
$105 \degrees$ $\dfrac {7 \pi} {12}$ Sine of $105 \degrees$ Cosine of $105 \degrees$ Tangent of $105 \degrees$ Cotangent of $105 \degrees$ Secant of $105 \degrees$ Cosecant of $105 \degrees$
$120 \degrees$ $\dfrac {2 \pi} 3$ Sine of $120 \degrees$ Cosine of $120 \degrees$ Tangent of $120 \degrees$ Cotangent of $120 \degrees$ Secant of $120 \degrees$ Cosecant of $120 \degrees$
$135 \degrees$ $\dfrac {3 \pi} 4$ Sine of $135 \degrees$ Cosine of $135 \degrees$ Tangent of $135 \degrees$ Cotangent of $135 \degrees$ Secant of $135 \degrees$ Cosecant of $135 \degrees$
$150 \degrees$ $\dfrac {5 \pi} 6$ Sine of $150 \degrees$ Cosine of $150 \degrees$ Tangent of $150 \degrees$ Cotangent of $150 \degrees$ Secant of $150 \degrees$ Cosecant of $150 \degrees$
$165 \degrees$ $\dfrac {11 \pi} {12}$ Sine of $165 \degrees$ Cosine of $165 \degrees$ Tangent of $165 \degrees$ Cotangent of $165 \degrees$ Secant of $165 \degrees$ Cosecant of $165 \degrees$
$180 \degrees$ $\pi$ Sine of $180 \degrees$ Cosine of $180 \degrees$ Tangent of $180 \degrees$ Cotangent of $180 \degrees$ Secant of $180 \degrees$ Cosecant of $180 \degrees$
$195 \degrees$ $\dfrac {13 \pi} {12}$ Sine of $195 \degrees$ Cosine of $195 \degrees$ Tangent of $195 \degrees$ Cotangent of $195 \degrees$ Secant of $195 \degrees$ Cosecant of $195 \degrees$
$210 \degrees$ $\dfrac {7 \pi} 6$ Sine of $210 \degrees$ Cosine of $210 \degrees$ Tangent of $210 \degrees$ Cotangent of $210 \degrees$ Secant of $210 \degrees$ Cosecant of $210 \degrees$
$225 \degrees$ $\dfrac {5 \pi} 4$ Sine of $225 \degrees$ Cosine of $225 \degrees$ Tangent of $225 \degrees$ Cotangent of $225 \degrees$ Secant of $225 \degrees$ Cosecant of $225 \degrees$
$240 \degrees$ $\dfrac {4 \pi} 3$ Sine of $240 \degrees$ Cosine of $240 \degrees$ Tangent of $240 \degrees$ Cotangent of $240 \degrees$ Secant of $240 \degrees$ Cosecant of $240 \degrees$
$255 \degrees$ $\dfrac {17 \pi} {12}$ Sine of $255 \degrees$ Cosine of $255 \degrees$ Tangent of $255 \degrees$ Cotangent of $255 \degrees$ Secant of $255 \degrees$ Cosecant of $255 \degrees$
$270 \degrees$ $\dfrac {3 \pi} 2$ Sine of $270 \degrees$ Cosine of $270 \degrees$ Tangent of $270 \degrees$ Cotangent of $270 \degrees$ Secant of $270 \degrees$ Cosecant of $270 \degrees$
$285 \degrees$ $\dfrac {19 \pi} {12}$ Sine of $285 \degrees$ Cosine of $285 \degrees$ Tangent of $285 \degrees$ Cotangent of $285 \degrees$ Secant of $285 \degrees$ Cosecant of $285 \degrees$
$300 \degrees$ $\dfrac {5 \pi} 3$ Sine of $300 \degrees$ Cosine of $300 \degrees$ Tangent of $300 \degrees$ Cotangent of $300 \degrees$ Secant of $300 \degrees$ Cosecant of $300 \degrees$
$315 \degrees$ $\dfrac {7 \pi} 4$ Sine of $315 \degrees$ Cosine of $315 \degrees$ Tangent of $315 \degrees$ Cotangent of $315 \degrees$ Secant of $315 \degrees$ Cosecant of $315 \degrees$
$330 \degrees$ $\dfrac {11 \pi} 6$ Sine of $330 \degrees$ Cosine of $330 \degrees$ Tangent of $330 \degrees$ Cotangent of $330 \degrees$ Secant of $330 \degrees$ Cosecant of $330 \degrees$
$345 \degrees$ $\dfrac {23 \pi} {12}$ Sine of $345 \degrees$ Cosine of $345 \degrees$ Tangent of $345 \degrees$ Cotangent of $345 \degrees$ Secant of $345 \degrees$ Cosecant of $345 \degrees$
$360 \degrees$ $2 \pi$ Sine of $360 \degrees$ Cosine of $360 \degrees$ Tangent of $360 \degrees$ Cotangent of $360 \degrees$ Secant of $360 \degrees$ Cosecant of $360 \degrees$

### Graphs of Trigonometric Functions

$5.22$ Graph of Sine Function
$5.23$ Graph of Cosine Function
$5.24$ Graph of Tangent Function
$5.25$ Graph of Cotangent Function
$5.26$ Graph of Secant Function
$5.27$ Graph of Cosecant Function

### Functions of Negative Angles

$5.28$ Sine Function is Odd
$5.29$ Cosine Function is Even
$5.30$ Tangent Function is Odd
$5.31$ Cosecant Function is Odd
$5.32$ Secant Function is Even
$5.33$ Cotangent Function is Odd

$5.34$ Sine of Sum
$5.34$ Sine of Difference
$5.35$ Cosine of Sum
$5.35$ Cosine of Difference
$5.36$ Tangent of Sum
$5.36$ Tangent of Difference
$5.37$ Cotangent of Sum
$5.37$ Cotangent of Difference

### Functions of Angles in All Quadrants in terms of those in Quadrant $\text I$

Trigonometric Functions of Negative Angle
Trigonometric Functions of Angle plus Right Angle
Trigonometric Functions of Complementary Angles
Trigonometric Functions of Angle plus Straight Angle
Trigonometric Functions of Supplementary Angles
Trigonometric Functions of Angle plus Three Right Angles
Trigonometric Functions of Three Right Angles less Angle
Trigonometric Functions of Angle plus Full Angle
Trigonometric Functions of Conjugate Angles

Not documented.

### Double Angle Formulas

$5.38$: Double Angle Formula for Sine
$5.39$: Double Angle Formula for Cosine
Double Angle Formula for Cosine: Corollary 2
Double Angle Formula for Cosine: Corollary 1
$5.40$: Double Angle Formula for Tangent

### Half Angle Formulas

$5.41$: Half Angle Formula for Sine
$5.42$: Half Angle Formula for Cosine
$5.43$: Half Angle Formula for Tangent
Half Angle Formula for Tangent: Corollary 1
Half Angle Formula for Tangent: Corollary 2
Half Angle Formula for Tangent: Corollary 3

### Multiple Angle Formulas

$5.44$: Triple Angle Formula for Sine
$5.45$: Triple Angle Formula for Cosine
$5.46$: Triple Angle Formula for Tangent
$5.47$: Quadruple Angle Formula for Sine
$5.48$: Quadruple Angle Formula for Cosine
$5.49$: Quadruple Angle Formula for Tangent
$5.50$: Quintuple Angle Formula for Sine
$5.51$: Quintuple Angle Formula for Cosine
$5.52$: Quintuple Angle Formula for Tangent

### Powers of Trigonometric Functions

$5.53$: Square of Sine
$5.54$: Square of Cosine
$5.55$: Cube of Sine
$5.56$: Cube of Cosine
$5.57$: Fourth Power of Sine
$5.58$: Fourth Power of Cosine
$5.59$: Fifth Power of Sine
$5.60$: Fifth Power of Cosine

### Sum, Difference and Product of Trigonometric Functions

$5.61$: Prosthaphaeresis Formula for Sine plus Sine
$5.62$: Prosthaphaeresis Formula for Sine minus Sine
$5.63$: Prosthaphaeresis Formula for Cosine plus Cosine
$5.64$: Prosthaphaeresis Formula for Cosine minus Cosine
$5.65$: Simpson's Formula for Sine by Sine
$5.66$: Simpson's Formula for Cosine by Cosine
$5.67$: Simpson's Formula for Sine by Cosine

### General Formulas

$5.68$: Sine of Integer Multiple of Argument
$5.69$: Cosine of Integer Multiple of Argument
$5.70$: Sine to Power of Odd Integer
$5.71$: Cosine to Power of Odd Integer
$5.72$: Sine to Power of Even Integer
$5.73$: Cosine to Power of Even Integer

### Principal Values for Inverse Trigonometric Functions

Real Arcsine
Real Arccosine
Real Arctangent
Real Arccotangent
Real Arcsecant
Real Arccosecant

### Relations between Inverse Trigonometric Functions

$5.74$: Sum of Arcsine and Arccosine
$5.75$: Sum of Arctangent and Arccotangent
$5.76$: Sum of Arcsecant and Arccosecant
$5.77$: Arcsine of Reciprocal equals Arccosecant
$5.78$: Arccosine of Reciprocal equals Arcsecant
$5.79$: Arctangent of Reciprocal equals Arccotangent
$5.80$: Arcsine is Odd Function
$5.81$: Arccosine of Negative Argument
$5.82$: Arctangent is Odd Function
$5.83$: Arccotangent of Negative Argument
$5.84$: Arcsecant of Negative Argument
$5.85$: Inverse Cosecant is Odd Function

### Graphs of Inverse Trigonometric Functions

$5.86$: Graph of Arcsine Function
$5.87$: Graph of Arccosine Function
$5.88$: Graph of Arctangent Function
$5.89$: Graph of Arccotangent Function
$5.90$: Graph of Arcsecant Function
$5.91$: Graph of Arccosecant Function

### Relationships between Sides and Angles of a Plane Triangle

$5.92$: Law of Sines
$5.93$: Law of Cosines
$5.94$: Law of Tangents
$5.95$: Sine of Angle of Triangle by Semiperimeter

### Relationships between Sides and Angles of a Spherical Triangle

$5.96$: Spherical Law of Sines
$5.97$: Spherical Law of Cosines
Spherical Law of Cosines for Angles
$5.98$: Spherical Law of Tangents
$5.99$: Cosine of Half Angle for Spherical Triangles
$5.100$: Cosine of Half Side for Spherical Triangles

### Napier's Rules for Right Angled Spherical Triangles

$5.101$: Napier's Tangent Rule for Right Spherical Triangles
$5.102$: Napier's Cosine Rule for Right Spherical Triangles

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