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

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 Plane


 * Cartesian Coordinate Pair:
 * $x$ Coordinate
 * $y$ Coordinate


 * Distance of Point from Origin in Cartesian Coordinates


 * Polar Coordinates:
 * Radial Coordinate
 * Angular Coordinate


 * Quadrants of Cartesian Coordinate Plane:
 * 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

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

Addition Formulas

 * $5.34$ 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

Relationships among Functions of Angles in Quadrant $\text I$

 * 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