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Topic: Hyperbolic cosine

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	The Hyperbolic Cosine Unfolding Quasi-Rasch Model (Site not responding. Last check: 2007-09-07)
		The Hyperbolic Cosine Model (HCM, Andrich and Luo, 1993) for dichotomous unfolding responses is derived from the Rasch model for 3-ordered-category responses, but is not, itself, a Rasch model.
		Andrich, D. A hyperbolic cosine latent trait model for unfolding polytomous responses: reconciling Thurstone and Likert methodologies.
		Andrich, D. and Luo, G. A hyperbolic cosine latent trait model for unfolding dichotomous single-stimulus responses.
	www.rasch.org /rmt/rmt161p.htm (367 words)

	Karl's Calculus Tutor - 11.5 Hyperbolic Substitution
		If hyperbolic substitution is not part of your curriculum, then skip ahead (unless you're curious) to Partial Fractions, and don't worry, we will cover the methods to integrate the functions that are covered here later on.
		Although the derivative of the hyperbolic sine is the hyperbolic cosine, the analogy does not quite stretch to finding the derivative of the hyperbolic cosine.
		The integrands that the hyperbolic functions address are all analogs of the ones that the trig functions were able to address.
	www.karlscalculus.org /hyper.html (2129 words)

	Hyperbolic functions (Site not responding. Last check: 2007-09-07)
		Hyperbolic functions are found useful in two main areas:
		On the Pickett N 4P-T, the hyperbolic functions are on the slide, not the stock.
		On the K and E this calculation is not quite so easy as it involves use of the C scale on the opposite face of the rule but similar principles apply.
	www.sliderules.clara.net /a-to-z/hyperbolic.htm (387 words)

	ANSI and GNU Common Lisp Document - sinh (Site not responding. Last check: 2007-09-07)
		The branch cut for the inverse hyperbolic sine function is in two pieces: one along the positive imaginary axis above i (inclusive), continuous with quadrant I, and one along the negative imaginary axis below -i (inclusive), continuous with quadrant III.
		The branch cut for the inverse hyperbolic tangent function is in two pieces: one along the negative real axis to the left of -1 (inclusive), continuous with quadrant III, and one along the positive real axis to the right of~1 (inclusive), continuous with quadrant I. The points -1 and~1 are excluded from the domain.
		Thus the range of the inverse hyperbolic tangent function is identical to that of the inverse hyperbolic sine function with the points -\pi i/2 and~\pi i/2 excluded.
	www.qucis.queensu.ca /software_docs/gnudev/gcl-ansi/gcl_767.html (523 words)

	Hyperbola - Wikipedia, the free encyclopedia
		For hyperbole, the figure of speech, see hyperbole.
		The rectangular hyperbola with the coordinate axes as its asymptotes is given by the equation xy=c, where c is a constant.
		Just as the sine and cosine functions give a parametric equation for the ellipse, so the hyperbolic sine and hyperbolic cosine give a parametric equation for the hyperbola.
	en.wikipedia.org /wiki/Hyperbola (431 words)

	Search Results for cosine - Encyclopædia Britannica
		the hyperbolic sine of z (written sinh z); the hyperbolic cosine of z (cosh z); the hyperbolic tangent of z (tanh z); and the hyperbolic cosecant, secant, and cotangent of z.
		Tutorial on the use of functions in oscillatory science with a focus on sine and cosine as applied in the study of waves.
		Discusses sines, cosines, tangents, trigonometric identities and functions, and the trigonometry of oblique and right-angled triangles.
	www.britannica.com /search?query=cosine&submit=Find&source=MWTEXT (447 words)

	Hyperbolic function : Hyperbolic cosine
		The hyperbolic functions are analogs of the ordinary trigonometric, or circular, functions.
		The parameter x can no longer be interpreted as an angle, though, and the hyperbolic functions are not periodic.
		The function cosh x is always positive, symmetric with respect to the y-axis and satisfies cosh 0 = 1, the function sinh x is symmetric with respect to the origin and hence sinh 0 = 0.
	www.fastload.org /hy/Hyperbolic_cosine.html (396 words)

	Xah: Special Plane Curves: Hyperbolic Trig Functions (Site not responding. Last check: 2007-09-07)
		They are called hyperbolic trig functions because they bear strong similarities to the trig functions.
		Trig functions relate to a circle, while hyperbolic trig functions relate to a rectangular hyperbola x^-y^==1 in a similar way.
		The hyperbolic cosine Cosh is famously known as catenary.
	www.xahlee.org /SpecialPlaneCurves_dir/Sinh_dir/sinh.html (158 words)

	timath.h
		cacosh calculates the hyperbolic area cosine w = acosh(z) of the complex number which real and imaginary parts are z_re and z_im, and stores real and imaginary part of the result in floating point destinations pointed to by w_re and w_im.
		ccosh calculates the hyperbolic cosine w = cosh(z) of the complex number which real and imaginary parts are z_re and z_im, and stores real and imaginary part of the result in floating point destinations pointed to by w_re and w_im.
		It calculates simultaneously the sine, the cosine and the tangent of the floating point value pointed to by xptr, and stores the results in floating point destinations pointed to by sine, cosine and tangent.
	tigcc.ticalc.org /doc/timath.html (7060 words)

	ipedia.com: Pythagorean theorem Article (Site not responding. Last check: 2007-09-07)
		There are two cases to consider -- spherical geometry and hyperbolic plane geometry; in each case, as in the Euclidean case, the result follows from the appropriate law of cosines:
		By using the Maclaurin series for the cosine function, it can be shown that as the radius R approaches infinity, the spherical form of the Pythagorean theorem approaches the Euclidean form.
		By using the Maclaurin series for this function, it can be shown that as a hyperbolic triangle becomes very small (i.e., as a, b, and c all approach zero), the hyperbolic form of the Pythagorean theorem approaches the Euclidean form.
	www.ipedia.com /pythagorean_theorem.html (1065 words)

	ComplexUtils (Math 1.2-dev API)
		Compute the hyperbolic cosine for the given complex argument.
		Compute the hyperbolic sine for the given complex argument.
		Compute the hyperbolic tangent for the given complex argument.
	jakarta.apache.org /commons/math/apidocs/org/apache/commons/math/complex/ComplexUtils.html (330 words)

	catenary
		Usually one adds a factor 1/2 to the formula, to form the hyperbolic cosine cosh(x) (or ch(x)).
		There are several ways to show that the chain has the form of the hyperbolic cosine.
		its inverse is the arc hyperbolic cosine arccosh(x)
	www.2dcurves.com /exponential/exponentialhc.html (473 words)

	tools.nb
		The cosine of q equals (cos t hyperbolic cosine absolute value of V, minus sine t hyperbolic sine of the absolute value of V times V normalized to V)
		The hyperbolic cosine of q equals (hyperbolic cos t cosine absolute value of V, hyperbolic sine t sine of the absolute value of V times V normalized to V)
		The hyperbolic arccosine of q equals the natural log of (q plus or minus the square root of q squared minus one).
	world.std.com /~sweetser/quaternions/intro/tools/s.html (981 words)

	Hyperbolic Trigonometric Functions
		The MacLaurin's series for the hyperbolic sine and cosine may be obtained from their definitions and the series for the exponential function.
		The MacLaurin's series formulae for the hyperbolic arc sine, arc cosine, arc cotangent, arc secant, arc cosecant, arc versed sine, arc coversed sine, and arc haversed sine are not interesting.
		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
	rism.com /Trig/hyperbol.htm (9675 words)

	Hyperbolic Fibonacci and Lucas Functions
		We can see that formulas (3) and (5) are similar with the hyperbolic sine (1) but formulas (4) and (6) are similar with the hyperbolic cosine (2).
		The theory of new classes of hyperbolic functions is stated in their article "Hyperbolic Fibonacci trigonometry" published in the "Reports of the Ukrainian Academy of Sciences" (1993, V. 208, No 7).
		Using the Fibonacci hyperbolic functions Bodnar proved that Fibonacci numbers arising at the surface of the phyllotaxis objects are a consequence of hyperbolic character of growth processes of the objects.
	www.goldenmuseum.com /0905HyperFibLuck_engl.html (596 words)

	Hyperbolic (JMSL Numerical Library)
		Pure Java implementation of the hyperbolic functions and their inverses.
		Returns the inverse hyperbolic cosine of its argument.
		value representing the number whose hyperbolic cosine is x.
	www.vni.com /products/imsl/jmsl/v30/api/com/imsl/math/Hyperbolic.html (191 words)

	Hyperbolic functions (Site not responding. Last check: 2007-09-07)
		Trigonometric and hyperbolic functions together with some other functions form a very important class of elementary functions, which are used in mathematics very often.
		Hyperbolic functions are used very widely for modeling of natural phenomena.
		Nikolay Lobatchevski used hyperbolic functions in your non-Euclidean geometry and therefore Lobatchevski's geometry is called hyperbolic geometry.
	www.goldenmuseum.com /0904HyperFunc_engl.html (281 words)

	Reference
		Return the arc sine, arc cosine, arc tangent of x, taking account of the specified units of angle
		Return the hyperbolic sine, hyperbolic cosine, hyperbolic tangent of x
		Return the inverse hyperbolic sine, inverse hyperbolic cosine, inverse hyperbolic tangent of x
	rcalc.sourceforge.net /doc/rcalc-reference.html (115 words)

	Karl's Calculus Tutor - Solution to Exercise 8.4-4
		The other way to do it is to run through the steps as you learned them in the main text.
		Do you expect that the hyperbolic cosine is an even function, an odd function, or neither?
		The hyperbolic cosine function is abbreviated by the notation,
	www.karlscalculus.org /pr8_4-4.html (294 words)

	MoreMath (Diamond Control Suite 3.0 API)
		Returns the inverse hyperbolic cosine of a number.
		Returns the inverse hyperbolic sine of a number.
		Returns the inverse hyperbolic tangent of a number.
	www.tvobjects.com /java/docs/diamondedge/util/MoreMath.html (622 words)

	CosH MathTrigComplex Function -- Entisoft Tools 2.0 Object Library (Site not responding. Last check: 2007-09-07)
		Hyperbolic cosine of a real or complex number.
		Cosine Function SinH Function TanH Function CscH Function ACosH Function HCos Function
		Function returns Null if vX is Null or cannot be fixed up to a real or complex number (as defined by the ComplexStringToReals function).
	www.entisoft.com /ESTools/MathTrigComplex_CosH.HTML (117 words)

	[No title]
		Press the appropriate number 1-6 to select the hyperbolic function of choice.
		Press OPTN to view the options menu, press F4 to select HYP and access the hyperbolic functions menu, then press the appropriate number 1-6 to select the hyperbolic function of choice.
		For example, we graph the hyperbolic sine function, sinh.
	www.humboldt.edu /~cmb2/technology/casiofx2indexfiles/hyperbolicfunctions.doc (198 words)

	Hyperbolic Cosine
		The Hyperbolic Cosine function calculates the hyperbolic cosine of a value in radians.
		The value of the hyperbolic cosine is equal to:
		Where e is a transcendental number that is the base of natural logarithms and is equal to approximately 2.71828183.
	www.tradingsolutions.com /functions/HyperbolicCosine.html (85 words)

	Hyperbolic Functions (Site not responding. Last check: 2007-09-07)
		Description: The acosh function returns the inverse hyperbolic cosine of x.
		Description: The sinhcosh function returns both the hyperbolic sine and hyperbolic cosine of x.
		These pages are expected to represent the University of Virginia community and the State of Virginia in a professional manner in accordance with the University of Virginia’s Computing Policies.
	www.itc.virginia.edu /research/intel/docs/c_ug/lm_hypr.htm (152 words)

	Maxima Manual - Trigonometric
		expands trigonometric and hyperbolic functions of sums of angles and of multiple angles occurring in exp. For best results, exp should be expanded.
		To enhance user control of simplification, this function expands only one level at a time, expanding sums of angles or multiple angles.
		default: [ALL] - controls the simplification of the composition of trig and hyperbolic functions with their inverse functions: If ALL, both e.g.
	www.ma.utexas.edu /maxima/maxima_14.html (707 words)

	Trigonometric Functions (Site not responding. Last check: 2007-09-07)
		Returns the element by element hyperbolic sine of the matrix.
		Returns the element by element hyperbolic cosine of the matrix.
		Returns the element by element hyperbolic tangent of the matrix.
	www.psatellite.com /products/html/matrixlib_api/a00021.html (1737 words)

	Search Results for hyperbolic - Encyclopædia Britannica
		The first description of hyperbolic geometry was given in the context of Euclid's postulates, and it was soon proved that all hyperbolic geometries differ only in scale (in the same sense that...
		Perhaps it was this desire for conceptual understanding that made Gauss reluctant to publish the fact that he was led more and more “to doubt the truth of geometry,” as he put it.
		Although the term is frequently used to refer only to hyperbolic geometry, common usage includes those few geometries (hyperbolic...
	www.britannica.com /search?query=hyperbolic&submit=Find&source=MWTEXT (471 words)

	Hyperbolic Functions
		The hyperbolic functions enjoy properties similar to the trigonometric functions; their definitions, though, are much more straightforward:
		The other hyperbolic functions are defined the same way, the rest of the trigonometric functions is defined:
		For every formula for the trigonometric functions, there is a similar (not necessary identical) formula for the hyperbolic functions:
	www.sosmath.com /trig/hyper/hyper01/hyper01.html (155 words)

	MathDouble (JSci API Documentation)
		Returns the arc hyperbolic cosine of a number.
		Returns the arc hyperbolic sine of a number.
		Returns the arc hyperbolic tangent of a number.
	jsci.sourceforge.net /api/JSci/maths/MathDouble.html (325 words)

	Inverse Hyperbolic Functions
		The hyperbolic sine function is a one-to-one function, and thus has an inverse.
		Note that the hyperbolic cosine function is not one-to-one, so let's restrict the domain to
		Here it is: Express the inverse hyperbolic cosine functions in terms of the logarithmic function!
	www.sosmath.com /trig/hyper/hyper03/hyper03.html (115 words)

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