The study of Angles and of the angular relationships of planar and 3-D figures is known as trigonometry.
The trigonometric functions (also called the Circular Functions) comprising trigonometry are the Cosecant
, Cosine , Cotangent , Secant , Sine , and
Tangent . The inverses of these functions are denoted , , ,
, , and . Note that the Notation here means Inverse
Function, *not* to the Power.

The trigonometric functions are most simply defined using the Unit Circle. Let be an Angle measured
counterclockwise from the *x*-Axis along an Arc of the Circle. Then is the horizontal
coordinate of the Arc endpoint, and is the vertical component. The Ratio
is defined as . As a result of this definition, the trigonometric functions are
periodic with period , so

(1) |

From the Pythagorean Theorem,

(2) |

(3) |

(4) |

(5) | |||

(6) | |||

(7) | |||

(8) | |||

(9) | |||

(10) |

Osborne's Rule gives a prescription for converting trigonometric identities to analogous identities for Hyperbolic Functions.

The Angles (with integers) for which the trigonometric function may be expressed in terms
of finite root extraction of *real numbers* are limited to values of which are precisely those which produce
constructible Polygons. Gauß showed these to be of the form

(11) |

(°) | (rad) | |||

0.0 | 0 | 0 | 1 | 0 |

15.0 | ||||

18.0 | ||||

22.5 | ||||

30.0 | ||||

36.0 | ||||

45.0 | 1 | |||

60.0 | ||||

90.0 | 1 | 0 | ||

180.0 | 0 | 0 |

The Inverse Trigonometric Functions are generally defined on the following domains.

Function | Domain |

or | |

or |

Inverse-forward identities are

(12) |

(13) |

(14) |

(15) |

(16) |

(17) |

(18) |

(19) |

Inverse sum identities include

(20) |

(21) |

(22) |

(23) |

Complex inverse identities in terms of Logarithms include

(24) | |||

(25) | |||

(26) | |||

(27) |

For Imaginary arguments,

(28) | |||

(29) |

For Complex arguments,

(30) | |||

(31) |

For the Absolute Square of Complex arguments ,

(32) | |||

(33) |

The Modulus also satisfies the curious identity

(34) |

(35) |

Trigonometric product formulas can be derived using the following figure (Kung 1996).

In the figure,

(36) | |||

(37) |

so

(38) | |||

(39) |

With and as previously defined, the above figure (Kung 1996) gives

(40) | |||

(41) |

Angle addition Formulas express trigonometric functions of sums of angles in terms of functions of and . They can be simply derived using Complex exponentials and the Euler Formula,

(42) |

(43) |

Taking the ratio gives the tangent angle addition Formula

(44) |

The angle addition Formulas can also be derived purely algebraically without the use of Complex Numbers. Consider the following figure.

From the large Right Triangle,

(45) | |||

(46) |

But, from the small triangle (inset at upper right),

(47) | |||

(48) |

Plugging and from (47) and (48) into (45) and (46) gives

(49) |

and

(50) |

Now solve (50) for ,

(51) |

(52) |

(53) |

so

(54) |

(55) |

(56) |

Multiplying out the Denominator gives

(57) |

so

(58) |

(59) |

(60) |

Summarizing (and explicitly writing out the identities for which is taken to be explicitly negative),

(61) | |||

(62) | |||

(63) | |||

(64) | |||

(65) | |||

(66) |

The sine and cosine angle addition identities can be summarized by the Matrix Equation

(67) |

The double angle formulas are

(68) | |||

(69) | |||

(70) | |||

(71) | |||

(72) |

General multiple angle formulas are

(73) | |

(74) | |

(75) | |

(76) | |

(77) |

(78) | |||

(79) | |||

(80) | |||

(81) | |||

(82) | |||

(83) |

Beyer (1987, p. 139) gives formulas up to .

Sum identities include

(84) |

Infinite sum identities include

(85) |

Trigonometric half-angle formulas include

(86) | |||

(87) | |||

(88) | |||

(89) | |||

(90) | |||

(91) |

The Prosthaphaeresis Formulas are

(92) | |||

(93) | |||

(94) | |||

(95) |

Related formulas are

(96) | |||

(97) | |||

(98) | |||

(99) |

Multiplying both sides by 2 gives the equations sometimes known as the Werner Formulas.

Trigonometric product/sum formulas are

(100) |

(101) |

Power formulas include

(102) | |||

(103) | |||

(104) |

and

(105) | |||

(106) | |||

(107) |

(Beyer 1987, p. 140). Formulas of these types can also be given analytically as

(108) | |||

(109) | |||

(110) | |||

(111) |

(Kogan), where is a Binomial Coefficient.

Trigonometric identities which prove useful in the construction of map projections include

(112) |

(113) | |||

(114) | |||

(115) | |||

(116) |

(117) |

(118) | |||

(119) | |||

(120) | |||

(121) |

(122) |

where

(123) | |||

(124) | |||

(125) | |||

(126) | |||

(127) |

(Snyder 1987).

**References**

Abramowitz, M. and Stegun, C. A. (Eds.). ``Circular Functions.'' §4.3 in
*Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing.*
New York: Dover, pp. 71-79, 1972.

Bahm, L. B. *The New Trigonometry on Your Own.* Patterson, NJ: Littlefield, Adams & Co., 1964.

Beyer, W. H. ``Trigonometry.'' *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press,
pp. 134-152, 1987.

Dixon, R. ``The Story of Sine and Cosine.'' § 4.4 in *Mathographics.* New York: Dover, pp. 102-106, 1991.

Hobson, E. W. *A Treatise on Plane Trigonometry.* London: Cambridge University Press, 1925.

Kells, L. M.; Kern, W. F.; and Bland, J. R. *Plane and Spherical Trigonometry.* New York: McGraw-Hill, 1940.

Kogan, S. ``A Note on Definite Integrals Involving Trigonometric Functions.'' http://www.mathsoft.com/asolve/constant/pi/sin/sin.html.

Kung, S. H. ``Proof Without Words: The Difference-Product Identities'' and
``Proof Without Words: The Sum-Product Identities.'' *Math. Mag.* **69**, 269, 1996.

Maor, E. *Trigonometric Delights.* Princeton, NJ: Princeton University Press, 1998.

Morrill, W. K. *Plane Trigonometry, rev. ed.* Dubuque, IA: Wm. C. Brown, 1964.

Robinson, R. M. ``A Curious Mathematical Identity.'' *Amer. Math. Monthly* **64**, 83-85, 1957.

Sloane, N. J. A. Sequence
A003401/M0505
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Snyder, J. P. *Map Projections--A Working Manual.* U. S. Geological Survey Professional Paper 1395.
Washington, DC: U. S. Government Printing Office, p. 19, 1987.

Thompson, J. E. *Trigonometry for the Practical Man.* Princeton, NJ: Van Nostrand.

Weisstein, E. W. ``Exact Values of Trigonometric Functions.'' Mathematica notebook TrigExact.m.

Yates, R. C. ``Trigonometric Functions.'' *A Handbook on Curves and Their Properties.* Ann Arbor, MI: J. W. Edwards, pp. 225-232, 1952.

Zill, D. G. and Dewar, J. M. *Trigonometry, 2nd ed.* New York: McGraw-Hill 1990.

© 1996-9

1999-05-26