en>SolidPhase: /* Common examples of F-tests */ link Statistical_model#Nested_models

2014-12-14T17:10:44Z

Common examples of F-tests: link Statistical_model#Nested_models

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en>Devadatta: /* External links */

2014-02-22T04:53:51Z

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92.78.126.138: /* One-way ANOVA example */

2014-01-16T16:23:24Z

One-way ANOVA example

← Older revision		Revision as of 18:23, 16 January 2014
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	~~PriceYou can get~~ a ~~high level~~ of ~~purchases~~ of ~~competitors in~~ the ~~neighborhood bookstore but that~~'~~s what it offers~~ a ~~veritable launchpad~~ of ~~miracle treatments: we could find he~~'s ~~couple gift baby shower pinterest met~~ his ~~match with one another~~. ~~Mr Angel~~ is ~~more digital~~. ~~This means~~ to ~~restructure your requirements~~ to ~~fit~~. ~~When you decide on your print jobs and opportunities~~, ~~businesses~~ of ~~all variable printing in~~ a ~~sense~~.<br><br>~~Feel free~~ to ~~visit my website~~: [~~http~~://~~www~~.~~pinterest.com~~/~~bottleyourbrand~~/~~tailgating~~-~~tips~~/ ~~follow~~ this ~~board on pinterest~~]		[[File:Triangle with notations 2.svg\|thumb\|198px\|right\|\|Figure 1 – A triangle. The angles ''α'', ''β'', and ''γ'' are respectively opposite the sides ''a'', ''b'', and ''c''.]]
			{{Trigonometry}}
			In [[trigonometry]], the '''law of tangents'''<ref>See [[Eli Maor]], ''Trigonometric Delights'', [[Princeton University Press]], 2002.</ref> is a statement about the relationship between the tangents of two angles of a [[triangle]] and the lengths of the opposing sides.

			In Figure 1, ''a'', ''b'', and ''c'' are the lengths of the three sides of the triangle, and α, β, and γ are the angles ''opposite'' those three respective sides. The law of [[trigonometric function\|tangent]]s states that

			:<math>\frac{a-b}{a+b} = \frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha+\beta)]}.</math>

			The law of tangents, although not as commonly known as the [[law of sines]] or the [[law of cosines]], is equivalent to the law of sines, and can be used in any case where two sides and the included angle, or two angles and a side, are known.

			The law of tangents for spherical triangles was described in the 13th century by [[Mathematics in medieval Islam\|Persian mathematician]] [[Nasir al-Din al-Tusi]] (1201–74), who also presented the law of sines for plane triangles in his five-volume work ''Treatise on the Quadrilateral''.<ref name=Debarnot>

			{{cite book \|url=http://books.google.com/books?id=cPGRYLlwbrEC&pg=PA182 \|title=Encyclopedia of the history of Arabic science, Volume 2 \|editor=Rushdī Rāshid, Régis Morelon \|author=Marie-Thérèse Debarnot \|chapter=Trigonometry \|isbn=0-415-12411-5 \|year=1996 \|page=182 \|publisher=Routledge}}

			</ref><ref name= Bosworth>

			{{cite book \|title=History of Civilizations of Central Asia, Volume 4, Part 2 \|editor=C. E. Bosworth, M.S.Asimov \|url=http://books.google.com/books?id=ELrRr0L8UOsC&pg=PA190 \|chapter=Trigonometry \|page=190 \|author=Q. Mushtaq, JL Berggren \|isbn=81-208-1596-3 \|year=2002 \|publisher=Motilal Banarsidass Publ.}}

			</ref>
			<!--
			==Applied to right triangles==

			The situation is simplified for right triangles.

			[[File:Triangle.Labels.svg]]

			Consider the right triangle shown. The angle at ''C'' is a right angle and the angle at ''A'' is θ (theta). The lengths of the sides of the triangle will be denoted as ''p'', ''q'' and ''r''.
			θ can be related to the lengths of the sides as follows:

			The sine of θ, written sin(θ), is defined as the ratio of the side opposite θ to the hypotenuse, that is, sin(θ) = ''r''/''q''.

			The cosine of θ, written cos(θ), is the side adjacent to θ over the hypotenuse, that is cos(θ) = ''p''/''q''.

			The tangent of θ, written tan(θ), is the ratio of the opposite side to the adjacent side, that is tan(θ) = ''r''/''p''.
			-->

			==Proof==
			To prove the law of tangents we can start with the [[law of sines]]:

			: <math>\frac{a}{\sin\alpha} = \frac{b}{\sin\beta}.</math>

			Let

			: <math>d = \frac{a}{\sin\alpha} = \frac{b}{\sin\beta},</math>

			so that

			: <math>a = d \sin\alpha \text{ and }b = d \sin\beta. \, </math>

			It follows that

			: <math>\frac{a-b}{a+b} = \frac{d \sin \alpha - d\sin\beta}{d\sin\alpha + d\sin\beta} = \frac{\sin \alpha - \sin\beta}{\sin\alpha + \sin\beta}.</math>

			Using the [[List of trigonometric identities#Product-to-sum and sum-to-product identities\|trigonometric identity]], the factor formula for sines specifically

			: <math> \sin(\alpha) \pm \sin(\beta) = 2 \sin\left( \frac{\alpha \pm \beta}{2} \right) \cos\left( \frac{\alpha \mp \beta}{2} \right), \;</math>

			we get

			:<math>\frac{a-b}{a+b} = \frac{2\sin\tfrac{1}{2}\left(\alpha-\beta\right)\cos\tfrac{1}{2}\left(\alpha+\beta\right)}{2\sin\tfrac{1}{2}\left(\alpha+\beta \right)\cos\tfrac{1}{2}\left(\alpha-\beta\right)} = \frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha+\beta)]}. \qquad\blacksquare</math>

			As an alternative to using the identity for the sum or difference of two sines, one may cite the trigonometric identity

			: <math> \tan\left( \frac{\alpha \pm \beta}{2} \right) = \frac{\sin\alpha \pm \sin\beta}{\cos\alpha + \cos\beta} </math>

			(see [[tangent half-angle formula]]).

			==Application==
			The law of tangents can be used to compute the missing side and angles of a triangle in which two sides <math>a,b</math> and the enclosed angle <math>\gamma</math> are given. From
			<math>\tan[\frac{1}{2}(\alpha-\beta)] = \frac{a-b}{a+b} \tan[\frac{1}{2}(\alpha+\beta)]=
			\frac{a-b}{a+b} \cot[\frac{\gamma}{2}]</math> one can compute <math>\alpha-\beta</math>; together with <math>\alpha+\beta=180^\circ-\gamma</math> this yields <math>\alpha</math> and <math>\beta</math>; the remaining side <math>c</math> can then be computed using the [[Law of sines]]. In the time before electronic calculators were available, this method
			was preferable to an application of the [[Law of cosines]] <math>c=\sqrt{a^2+b^2-2ab \cos \gamma}</math>, as this latter law necessitated an additional lookups in a [[logarithm table]], in order to compute the square root. In modern times the law of tangents may have better [[numerical analysis\|numerical]] properties than the law of cosines: If <math>\gamma</math> is small, and <math>a</math> and <math>b</math> are almost equal, then an application of the law of cosines leads to a subtraction of almost equal values, which implies a [[Loss of significance\|loss of significant digits]].

			==See also==
			* [[Law of sines]]
			* [[Law of cosines]]
			* [[Mollweide's formula]]
			* [[Half-side formula]]

			== Notes ==
			{{reflist}}

			[[Category:Trigonometry]]
			[[Category:Articles containing proofs]]
			[[Category:Theorems in plane geometry]]

2001:3C8:5007:FF18:2864:4710:BE5F:4542: /* Common examples of F-tests */

2012-08-23T06:33:36Z

Common examples of F-tests

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F-test - Revision history

en>SolidPhase: /* Common examples of F-tests */ link Statistical_model#Nested_models

en>Devadatta: /* External links */

92.78.126.138: /* One-way ANOVA example */

2001:3C8:5007:FF18:2864:4710:BE5F:4542: /* Common examples of F-tests */