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Topic: Exact trigonometric constants

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	Trigonometric function (Site not responding. Last check: 2007-09-11)
		The Computation of trigonometric functions is a complicated subject, which can today be avoided by most people because of the widespread availability of computers and scientific calculators that provide built-in trigonometric functions for any angle.
		The six trigonometric functions can also be defined in terms of the unit circle, the circle of radius one centered at the origin.
		The earliest systematic study of trigonometric functions and tabulation of their values was performed by Hipparchus of Nicaea (180-125 BC), who tabulated the lengths of circle arcs (angle A times radius r) with the lengths of the subtending chords (2''r'' sin(''A''/2)).
	trigonometric-function.iqnaut.net (3366 words)

	Encyclopedia :: encyclopedia : Exact trigonometric constants (Site not responding. Last check: 2007-09-11)
		Exact constant expressions for trigonometric expressions are sometimes useful, mainly for simplifying solutions into radical forms which allow further simplification.
		The derivation of sine, cosine, and tangent constants into radial forms is based upon the constructability of right triangles.
		Trigonometric functions of angles 0° to 90° by degree
	www.hallencyclopedia.com /topic/Exact_trigonometric_constants.html (667 words)

	Survey Manual Version January 2000 (Site not responding. Last check: 2007-09-11)
		Although they are known to be not exact, higher order NGS control points are deemed of sufficient accuracy to be the control for all other less exact surveys.
		Improper Prisms or Preset Prism Constants Most EDM prisms have built in cross lens constants of 30 millimeters (0.098 feet). Each EDM manufacturer provides for their instruments' direct measurements by the combination of an internal adjustment within the instrument, and position of the prism in relation to the vertical axis of the prism.
		Trigonometric vertical measurement is a procedure whereby vertical differences in elevation are computed from slope distance and zenith angle measurements.
	www.state.nj.us /transportation/eng/documents/survey/Chapter3.htm (7130 words)

	Common Terms in Mathematics [Dilara & M.Tevfik Dorak]
		A state of disorder and irregularity whose evolution in time, though governed by simple exact laws, is highly sensitive to starting conditions: a small variation in these conditions will produce wildly different results, so that long-term behaviour of chaotic systems cannot be predicted.
		Trigonometry: The branch of mathematics that is concerned with the trigonometric functions.
		Trigonometric identities are the results that hold true for all angles.
	members.tripod.com /~dorakd/dilara/glosmath.html (3948 words)

	Generating trigonometric tables - Wikipedia, the free encyclopedia
		Interpolation of simple look-up tables of trigonometric functions are still used in computer graphics, where accurate calculations are either not needed, or cannot be made fast enough.
		Another important application of trigonometric tables and generation schemes is for fast Fourier transform (FFT) algorithms, where the same trigonometric function values (called twiddle factors) must be evaluated many times in a given transform, especially in the common case where many transforms of the same size are computed.
		Historically, the earliest method by which trigonometric tables were computed, and probably the most common until the advent of computers, was to repeatedly apply the half-angle and angle-addition trigonometric identities starting from a known value (such as sin(π/2)=1, cos(π/2)=0).
	en.wikipedia.org /wiki/Generating_trigonometric_tables (738 words)

	Journal of Theoretics -4D Unified Math, A Unified Theory
		Though the constants will represent known arithmetic values they are not to be calculated as mathematical functions during the process.
		The distance (as in the measure of length) of the speed of light is a variable (relative) where the speed of light itself is a constant.
		In 4D Math the constant of π is equal to the geometric shape of the inlet and outlet portals of the TIME tunnels created by the 918 paired photons.
	www.journaloftheoretics.com /Articles/3-1/4D-final.htm (5903 words)

	Quantities and Numbers
		An implementation shall support exact integers throughout the range of numbers that may be used for indexes of lists and strings or that may result from computing the length of a list or string.
		If one of these procedures is unable to deliver an exact result when given exact arguments, then it may either report a violation of an implementation restriction or it may silently coerce its result to an inexact number.
		Some examples also assume that certain numerical constants written using an inexact notation may be represented without loss of accuracy; the inexact constants were chosen so that this is likely to be true in implementations that use floating point numbers to represent inexact quantities.
	www.jclark.com /dsssl/IS/dsssl49.htm (2479 words)

	Hyperbolic Trigonometric Functions
		These constant coefficients a, b, c, d, e, and f are not related to the previous constants a, b, c, and e.
		Newton showed that this proportionality constant, in the third law, is (2 pi / sqrt(G M)), where G is the gravitational constant and M = ms^3 / (mp + ms)^2, where ms is the mass of the sun and mp is the mass of the planet.
		The values of the inverse hyperbolic trigonometric functions have to be obtained from the foregoing arctangent, by solving the quadratic equations of the identities and
	www.rism.com /Trig/hyperbol.htm (9845 words)

	Numerical Calculations Using Mathematica
		Note that this computation does not yield the exact value of 0, although the significance of this inaccuracy may be considered as rather small.
		One of the constants is the mathematical constant
		Note that this constant is irrational and cannot by held in terms of a decimal expansion by any computer.
	www.mathcs.emory.edu /ccs/math190g/ModuleI/ModIP2.html (1037 words)

	Trigonometry and Basic Functions - Numericana
		Modern usage is to consider only the 3 preferred trigonometric functions (sine, cosine and tangent) whereas their 3 reciprocals (cosecant, secant and cotangent) are being deprecated.
		Each of those 6 trigonometric functions is the ratio of two sides in a right triangle where one of the acute angles is specified.
		The better approach, whenever possible, is to compute the exact Chebyshev expansion of the target function and to truncate that in order to obtain a good approximation by a polynomial of low degree...
	home.att.net /~numericana/answer/functions.htm (4111 words)

	MATH2070: LAB 9: Legendre Polynomials and L2 Approximation (Site not responding. Last check: 2007-09-11)
		The third selection is the trigonometric functions, and the final selection is a set of piecewise constant functions.
		Approximation by piecewise constants is not subject to error until ridiculously large numbers of pieces are employed.
		When you used trigonometric polynomial interpolation in Lab 7, you looked at the error for a sawtooth function and saw the Gibb's phenomenon, which kept the error from going to zero.
	www.math.pitt.edu /~sussmanm/2070Fall06/lab_09/lab_09.html (3473 words)

	Mathematics Placement Exam Algebra Review Material s
		Values of the trigonometric functions at 0°, 30°, 45°, 60° and 90° should be memorized.
		Given the quadrant of an angle and the value of one of its trigonometric functions, we can use the preceding identities to find the values of the other trigonometric functions.
		Any trigonometric expression can be written entirely in terms of the sine and cosine functions by using the reciprocal and quotient identities.
	www.math.colostate.edu /dept/IMP/SG/MPE_Review/trigonometry.html (2869 words)

	pop11 ref numbers
		Otherwise, when the function is irrational, or one or more of the arguments is floating-point, all arguments are converted to double-length floating-point and the computation performed with double-float arithmetic (this is the rule of 'floating-point contagion').
		In both cases, the units of the angle (radians or degrees) are controlled by the boolean variable popradians (this applies equally when the angle is complex).
		pop_plus_epsilon_decimal [constant] pop_plus_epsilon_ddecimal [constant] These are the smallest positive numbers in each format which when added to 1.0 of the same format produce a value not equal to 1.0.
	openpoplog.sourceforge.net /ref/numbers.html (5123 words)

	Implications of a Non-Constant Velocity of Light
		Measurements on constants of physics which do not carry dimensions of time (seconds or 1/seconds; or powers thereof) are found to be truly fixed and invariant.
		If the "universal constants" are different at the position of supernova 1987A, for example, then the physics is different and an observer in that frame should be able to determine that he is in a unique position relative to any other frame of reference and vice-versa.
		The exact change in light-speed per quantum change is somewhat more indefinite, being dependent upon what value is placed on the Hubble constant.) Note also that light-speed does not itself jump down a quantum number as it is a smooth function.
	ldolphin.org /cdkconseq.html (22804 words)

	Karl's Calculus Tutor - 11.5 Trigonometric Substitution
		Your Japanese translator at the other end will certainly recognize the words by their phonetic spellings, so he shall be able to transcribe them back into Japanese characters that my mother can read.
		Trigonometric substitution is a big topic because there are lots of variations.
		Here is an integrand that you've seen before because it's on the table, but it is useful to see how we can use a trigonometric substitution on it.
	www.karlscalculus.org /calc11_4.html (2745 words)

	ANSI and GNU Common Lisp Document - Numbers
		When only number is given, the two results are exact; the mathematical sum of the two results is always equal to the mathematical value of number.
		The identity and nature of the values of each of these variables is implementation-dependent, except that it must be distinct from each of the values of the others, and it must be a valid first argument to the function boole.
		Each of these constants has a value which is one of the sixteen possible bit-wise logical operation specifiers.
	ugweb.cs.ualberta.ca /~c325/gcl/gcl_12.html (10184 words)

	Approximation of Functions
		To do an exact fit, at each grid point we equate the value of the function at that grid point to the value of the approximation at the same grid point:
		Naturally, if the function itself is not a finite power or trigonometric series, then we need more and more terms for more and more accuracy.
		A power or trigonometric series with infinite number of terms may yield the exact value of the function, as is the case with Taylor and Fourier series.
	www.numericalmathematics.com /approximation_of_functions.htm (499 words)

	Common Errors in College Math
		But the beginner writes neither phrase, and the reader is expected to guess which it is. This is like a novel in which there are many flashbacks and also much foreshadowing, but all the verbs are in present tense; the reader must try to figure out a logical order in which the events actually occur.
		In particular, the constant function 1 is the function given by p(x)=1 for all x.
		Handling constants of integration gets even more complicated in the first course on differential equations, and there are even more kinds of errors possible.
	www.math.vanderbilt.edu /~schectex/commerrs (13343 words)

	Numerical and Related Functions
		Exponential and trigonometric functions use Taylor series, stable recursion by argument doubling, and functional relations.
		Log and inverse trigonometric functions use Taylor series and functional relations.
		For exact integer inputs, enough digits are computed to deduce exact integer results.
	documents.wolfram.com /teachersedition/MathematicaBook/A.9.4.html (948 words)

	Exact trigonometric constants - Wikipedia, the free encyclopedia
		First, it is always possible to apply a half-angle formula and find an exact expression for the cosine of one-half the smallest angle on the list.
		In practice, all values of sine, cosine, and tangent not found in this article are approximated using the techniques described at Generating trigonometric tables.
		As an example of the use of these constants, consider a dodecahedron with the following volume, where e is the length of an edge:
	en.wikipedia.org /wiki/Exact_trigonometric_constants (787 words)

	University of Alabama Undergraduate Catalog 2006-08
		The course includes study of trigonometric functions, inverse trigonometric functions, trigonometric identities, and trigonometric equations.
		Properties and graphs of exponential, logarithmic, and trigonometric functions are emphasized.
		Topics include the limit of a function; the derivative of algebraic, trigonometric, exponential, and logarithmic functions; and the definite integral.
	www.ua.edu /catalogs/undergraduate/45400.html (2479 words)

	Information Technology Services (Site not responding. Last check: 2007-09-11)
		One of the important Maple properties is the ability to do exact arithmetic: rational numbers are not reduced to float-point approximation.
		The commonly used numerical functions and constants are shown in the following Table 2 and Table 3.
		It can be used to solve a set of equations and gives exact solutions to equations with rational coefficients.
	www.utexas.edu /its/rc/tutorials/maple (3775 words)

	Quantities and Numbers
		procedures shall return an exact integer, and it shall be an error to use anything but an exact integer as an index.
		, when applied to exact arguments, to produce exact answers whenever possible (for example the square root of an exact 4 ought to be an exact 2).
		is used to compare quantities of mixed exactness, and the numerical value of the result cannot be represented as an inexact quantity without loss of accuracy, then the procedure may report a violation of an implementation restriction.
	www.cs.berkeley.edu /~wilensky/CS294/dsssl/html/h3-29.htm (2318 words)

	Roanoke Valley Governor's School
		F8-6 Given a real-world situation in which one variable is proportional to a non-integer power of another variable, determine the proportionality constant, and use the resulting variation function as a math model.
		COMPETENCY 6: Use trigonometric and circular identities to transform and solve trigonometric and circular equations.
		6-2 Given a trigonometric equation, prove that it is an identity.
	rvgs.k12.va.us /courses/cpcomp.htm (2219 words)

	FCPS Instructional Services
		The student will use the definitions of trigonometric functions to find the sine, cosine, tangent, cotangent, secant, and cosecant of an angle.
		The student will identify the graph of trigonometric functions and determine their period, amplitude, and phase shift.
		Change points and equations from polar to rectangular form and vice versa, classify polar equations, predict the effects of changes in the constants, and find points of intersections of polar graphs.
	www.fcps.k12.va.us /DIS/pos/math/trigonometry315000.htm (1125 words)

	PHASER Module: Archimedes' Pi MAP
		The semi-perimeters are iteratively approximated using trigonometric identities which Archimedes derives.
		The process of subdivision halves at each step the angle φ formed by a polygon vertex, the center of the circle and the midpoint of a polygon face incident on the vertex.
		Archimedes did not use the notation of trigonometric functions.
	www.phaser.com /modules/historic/archimedes/index.html (1045 words)

	Mathematica Tutorials
		If all of the constants in an algebraic expression are numbers, the expression is called an algebraic number.
		In this particular example there is only one solution, and that solution involves a rule for only one function, but this same form of result can also be used to represent results that involve more than one solution or rules for more than one function.
		In this example the denominators happen to cancel with factors from the numerator, leaving a result that is significantly simpler than the original expression.
	library.wolfram.com /conferences/devconf99/withoff/Basic3.html (2457 words)

	SparkNotes: Trigonometric Functions: Terms and Formulae
		Periodic Function - A function that repeats itself in regular intervals; it follows this equation: f (x + c) = f (x), where c is a constant.
		Rule - The part of a function that dictates the exact correspondence between the elements of one set and the elements of another set.
		Trigonometric Functions - There are six trigonometric functions: Sine, cosine, tangent, cosecant, secant, and cotangent.
	www.sparknotes.com /math/trigonometry/trigonometricfunctions/terms.html (343 words)

	Trapping and Wiggling: Elastohydrodynamics of Driven Microfilaments -- Wiggins et al. 74 (2): 1043 -- Biophysical ...
		This is the exact solution of the linearized homogeneous equation.
		in time, except for the zero (average) mode, which is constant.
		The exact solution to EHDII given in Appendix C can be used to calculate the function
	www.biophysj.org /cgi/content/full/74/2/1043 (4399 words)

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