| 释义 |
TrigonometryThe 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  , andTangent  . The inverses of these functions are denoted  ,  ,  , ,  , and  . Note that the  Notation here means InverseFunction, not  to the  Power.
The trigonometric functions are most simply defined using the Unit Circle. Let  be an Angle measuredcounterclockwise from the x-Axis along an Arc of the Circle. Then  is the horizontalcoordinate of the Arc endpoint, and  is the vertical component. The Ratio is defined as  . As a result of this definition, the trigonometric functions areperiodic with period  , so
 | (1) |
 is an Integer and func is a trigonometric function.
From the Pythagorean Theorem,
 | (2) |
 | (3) |
 | (4) |
Osborne's Rule gives a prescription for converting trigonometric identities to analogous identities forHyperbolic Functions.
The Angles  (with  integers) for which the trigonometric function may be expressed in termsof finite root extraction of real numbers are limited to values of  which are precisely those which produceconstructible Gauß  showed these to be of the form
 | (11) |
 is an Integer  and the  are distinct Fermat Primes. The first fewvalues are  , 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, ... (Sloane's A003401). Although formulas for trigonometricfunctions may be found analytically for other as well, the expressions involve Roots of Complex Numbers obtained by solving a Cubic, Quartic, or higherorder equation. The cases  and  involve the Cubic Equation and Quartic Equation, respectively.A partial table of the analytic values of Sine, Cosine, and Tangent for arguments  is given below. Derivations of these formulas appear in the following entries.
 (°) | (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. 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
For Imaginary arguments,
For Complex arguments,
For the Absolute Square of Complex arguments  ,
The Modulus also satisfies the curious identity
 | (34) |
 | (35) |
 ,  , and  (Robinson 1957).
Trigonometric product formulas can be derived using the following figure (Kung 1996). In the figure,
so
With and  as previously defined, the above figure (Kung 1996) gives
Angle addition Formulas express trigonometric functions of sums of angles  in terms of functions of  and  . They can be simply derived using Complexexponentials and the Euler Formula,
Taking the ratio gives the tangent angle addition Formula
The angle addition Formulas can also be derived purely algebraically without the use ofComplex Numbers. Consider the following figure. From the large Right Triangle,
But, from the small triangle (inset at upper right),
Plugging  and  from (47) and (48) into (45) and (46) gives
and
Now solve (50) for  ,
 | (51) |
 | (52) |
so
 | (54) |
 | (55) |
Multiplying out the Denominator givesso
 | (58) |
| (59) |
Summarizing (and explicitly writing out the identities for which is taken to be explicitly negative),
The sine and cosine angle addition identities can be summarized by the Matrix Equation
 | (67) |
The double angle formulas are
General multiple angle formulas are Therefore, any trigonometric function of a sum can be broken up into a sum of trigonometric functions with  cross terms. Particular cases for multiple angle formulas up to  are given below.
Beyer (1987, p. 139) gives formulas up to  .
Sum identities include
Infinite sum identities include
 | (85) |
Trigonometric half-angle formulas include
 |  |  | (86) |  | |  | (87) |  | |  | (88) | | | |  | (89) | | | |  | (90) | | | |  | (91) |
The Prosthaphaeresis Formulas are
Related formulas are
Multiplying both sides by 2 gives the equations sometimes known as the Werner Formulas.
Trigonometric product/sum formulas are
 | (100) |
 | (101) |
Power formulas include
and
(Beyer 1987, p. 140). Formulas of these types can also be given analytically as
(Kogan), where  is a Binomial Coefficient.
Trigonometric identities which prove useful in the construction of map projections include  | |  | (112) |
where
 | |  | (117) |
where
 | |  | (122) |
where
 |  |  | (123) |  | |  | (124) |  | |  | (125) |  | |  | (126) |  | |  | (127) |
(Snyder 1987).See also Cosecant, Cosine, Cotangent, Euclidean Number, Inverse Cosecant, InverseCosine, Inverse Cotangent, Inverse Secant, Inverse Sine, Inverse Tangent, InverseTrigonometric Functions, Osborne's Rule, Polygon, Secant, Sine, Tangent,Trigonometry Values: Pi, Trigonometry Values: Pi/2, Trigonometry Values: Pi/3, Trigonometry Values: Pi/4, Trigonometry Values: Pi/5, Trigonometry Values: Pi/6, Trigonometry Values: Pi/7, Trigonometry Values: Pi/8, Trigonometry Values: Pi/9, Trigonometry Values: Pi/10, Trigonometry Values: Pi/11, Trigonometry Values: Pi/12, Trigonometry Values: Pi/15, Trigonometry Values: Pi/16, Trigonometry Values: Pi/17, Trigonometry Values: Pi/18, Trigonometry Values: Pi/20, Trigonometry Values 0, Werner Formulas
References
 TrigonometryAbramowitz, 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. SequenceA003401/M0505in ``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.
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