Tangent function - Revision history

Vipul: /* Power series and Taylor series */

2012-07-08T14:43:53Z

Power series and Taylor series

← Older revision		Revision as of 14:43, 8 July 2012
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	<math>\sum_{k=0}^\infty \frac{A_{2k+1}x^{2k+1}}{(2k + 1)!}</math>		<math>\sum_{k=0}^\infty \frac{A_{2k+1}x^{2k+1}}{(2k + 1)!}</math>

			===Taylor series equals power series===

			The Taylor series for <math>\tan</math> converges to <math>\tan</math> on its [[interval of convergence]], which is <math>(-\pi/2,\pi/2)</math>.

	==Limit computations==		==Limit computations==

Vipul: /* Power series and Taylor series */

2012-07-08T14:41:36Z

Power series and Taylor series

← Older revision		Revision as of 14:41, 8 July 2012
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	==Power series and Taylor series==		==Power series and Taylor series==

	The <math>\tan</math> function has a [[Taylor series]] at <math>x = 0</math>~~. These is no clean expression for the general term of the series~~. The first few terms are indicated below:		===Computation of Taylor series===

			The <math>\tan</math> function has a [[Taylor series]] at <math>x = 0</math>. The first few terms are indicated below:

	<math>\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \dots</math>		<math>\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \dots</math>
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	* All the terms with nonzero coefficients are odd degree terms, because <math>\tan</math> is an [[odd function]].		* All the terms with nonzero coefficients are odd degree terms, because <math>\tan</math> is an [[odd function]].
	* All the coefficients on odd degree terms are positive. This follows from the fact that all the higher derivatives of <math>\tan</math>, evaluated at 0, are positive.		* All the coefficients on odd degree terms are positive. This follows from the fact that all the higher derivatives of <math>\tan</math>, evaluated at 0, are positive.

			The coefficients can be written explicitly using the odd-degree [[up/down number]]s, which are also called the '''tangent numbers''' or '''zag numbers'''. Explicitly, if <math>A_n</math> denotes the <math>n^{th}</math> tangent number, then the Taylor series for <math>\tan</math> is:

			<math>\sum_{k=0}^\infty \frac{A_{2k+1}x^{2k+1}}{(2k + 1)!}</math>

	==Limit computations==		==Limit computations==

Vipul: /* Order of zero */

2011-12-19T17:41:05Z

Order of zero

← Older revision		Revision as of 17:41, 19 December 2011
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	\| Simple manipulation, using <math>\lim_{x \to 0} (\sin x)/x = 1</math> \|\| <math>\! \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sin x}{x} \frac{1}{\cos x} = \lim_{x \to 0} \frac{\sin x}{x} \lim_{x \to 0} \frac{1}{\cos x} = 1 \cdot 1 = 1</math>		\| Simple manipulation, using <math>\lim_{x \to 0} (\sin x)/x = 1</math> \|\| <math>\! \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sin x}{x} \frac{1}{\cos x} = \lim_{x \to 0} \frac{\sin x}{x} \lim_{x \to 0} \frac{1}{\cos x} = 1 \cdot 1 = 1</math>
	\|-		\|-
	\| Using the [[L'Hopital rule]] \|\| <math>\lim_{x \to 0} \frac{\tan x}{x} \~~stacrek~~{*}{=} \lim_{x \to 0} \frac{\sec^2x}{1} = \sec^20 = 1</math>		\| Using the [[L'Hopital rule]] \|\| <math>\lim_{x \to 0} \frac{\tan x}{x} \stackrel{*}{=} \lim_{x \to 0} \frac{\sec^2x}{1} = \sec^20 = 1</math>
	\|-		\|-
	\| Using the power series \|\| We have <math>\tan x = x + \frac{x^3}{3} + O(x^5)</math>, so <math>(\tan x)/x = 1 + \frac{x^2}{3} + O(x^4)</math>. Taking the limit at <math>x = 0</math> gives 1.		\| Using the power series \|\| We have <math>\tan x = x + \frac{x^3}{3} + O(x^5)</math>, so <math>(\tan x)/x = 1 + \frac{x^2}{3} + O(x^4)</math>. Taking the limit at <math>x = 0</math> gives 1.
	\|}		\|}

Vipul: /* Power series and Taylor series */

2011-12-19T17:40:37Z

Power series and Taylor series

← Older revision		Revision as of 17:40, 19 December 2011
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	* All the terms with nonzero coefficients are odd degree terms, because <math>\tan</math> is an [[odd function]].		* All the terms with nonzero coefficients are odd degree terms, because <math>\tan</math> is an [[odd function]].
	* All the coefficients on odd degree terms are positive. This follows from the fact that all the higher derivatives of <math>\tan</math>, evaluated at 0, are positive.		* All the coefficients on odd degree terms are positive. This follows from the fact that all the higher derivatives of <math>\tan</math>, evaluated at 0, are positive.

			==Limit computations==

			===Order of zero===

			We get the following limit from the power series:

			<math>\lim_{x \to 0} \frac{\tan x}{x} = 1</math>

			Thus, the order of the zero of <math>\tan</math> at 0 is 1 and the residue is 1.

			The limit can be computed in many ways:

			{\| class="sortable" border="1"
			! Name of method for computing the limit !! Details
			\|-
			\| Simple manipulation, using <math>\lim_{x \to 0} (\sin x)/x = 1</math> \|\| <math>\! \lim_{x \to 0} \frac{\tan x}{x} = \lim_{x \to 0} \frac{\sin x}{x} \frac{1}{\cos x} = \lim_{x \to 0} \frac{\sin x}{x} \lim_{x \to 0} \frac{1}{\cos x} = 1 \cdot 1 = 1</math>
			\|-
			\| Using the [[L'Hopital rule]] \|\| <math>\lim_{x \to 0} \frac{\tan x}{x} \stacrek{*}{=} \lim_{x \to 0} \frac{\sec^2x}{1} = \sec^20 = 1</math>
			\|-
			\| Using the power series \|\| We have <math>\tan x = x + \frac{x^3}{3} + O(x^5)</math>, so <math>(\tan x)/x = 1 + \frac{x^2}{3} + O(x^4)</math>. Taking the limit at <math>x = 0</math> gives 1.
			\|}

Vipul: /* Differential equations */

2011-12-19T17:37:19Z

Differential equations

← Older revision		Revision as of 17:37, 19 December 2011
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	==Differential equations==		==Differential equations==

	===Autonomous differential equations with this function as solution:		===Autonomous differential equations with this function as solution===

	The standard differential equation is:		The standard differential equation is:
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	The general solution to this is <math>y= \tan(x + \varphi)</math> where <math>\varphi</math> is a constant. The value of <math>\varphi</math> can be determined using initial value conditions.		The general solution to this is <math>y= \tan(x + \varphi)</math> where <math>\varphi</math> is a constant. The value of <math>\varphi</math> can be determined using initial value conditions.

			==Power series and Taylor series==

			The <math>\tan</math> function has a [[Taylor series]] at <math>x = 0</math>. These is no clean expression for the general term of the series. The first few terms are indicated below:

			<math>\tan x = x + \frac{x^3}{3} + \frac{2x^5}{15} + \frac{17x^7}{315} + \dots</math>

			Note the following:

			* All the terms with nonzero coefficients are odd degree terms, because <math>\tan</math> is an [[odd function]].
			* All the coefficients on odd degree terms are positive. This follows from the fact that all the higher derivatives of <math>\tan</math>, evaluated at 0, are positive.

Vipul: /* Higher antiderivatives */

2011-12-19T17:33:29Z

Higher antiderivatives

← Older revision		Revision as of 17:33, 19 December 2011
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	It is '''not''' possible to antidifferentiate the first antiderivative of <math>\tan</math> within the universe of elementarily expressible functions.		It is '''not''' possible to antidifferentiate the first antiderivative of <math>\tan</math> within the universe of elementarily expressible functions.

			==Differential equations==

			===Autonomous differential equations with this function as solution:

			The standard differential equation is:

			<math>\! \frac{dy}{dx} = 1 +y^2</math>

			The general solution to this is <math>y= \tan(x + \varphi)</math> where <math>\varphi</math> is a constant. The value of <math>\varphi</math> can be determined using initial value conditions.

Vipul: /* Values at some acute angles */

2011-12-19T17:31:18Z

Values at some acute angles

← Older revision		Revision as of 17:31, 19 December 2011
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	\| <math>\pi/2</math> \|\| 1.5708 \|\| <math>90\,^\circ</math> \|\| undefined \|\| undefined		\| <math>\pi/2</math> \|\| 1.5708 \|\| <math>90\,^\circ</math> \|\| undefined \|\| undefined
	\|}		\|}

			Note that for <math>x</math> close to zero, <math>\tan x</math> is just slightly bigger than <math>x</math>. As <math>x</math> becomes bigger, the functions diverge and as <math>x \to \pi/2^-</math>, <math>\tan x \to +\infty</math>.

	==Differentiation==		==Differentiation==

Vipul: /* Identities */

2011-12-19T17:28:29Z

Identities

← Older revision		Revision as of 17:28, 19 December 2011
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	\| formula for tangent of acute angle in terms of double angle cosine \|\| <math>\tan x = \sqrt{(1 - \cos (2x))/(1 + \cos (2x))}</math>		\| formula for tangent of acute angle in terms of double angle cosine \|\| <math>\tan x = \sqrt{(1 - \cos (2x))/(1 + \cos (2x))}</math>
	\|-		\|-
	\| other symmetries \|\| periodicity: <math>\tan(\pi + x) = \tan x</math><br>\tan(\pi - x) = -\tan x</math><br>[[odd function]]: <math>\tan(-x) = -\tan x</math>		\| other symmetries \|\| periodicity: <math>\tan(\pi + x) = \tan x</math><br><math>\tan(\pi - x) = -\tan x</math><br>[[odd function]]: <math>\tan(-x) = -\tan x</math>
	\|}		\|}

	==Values==		==Values==

Vipul: /* Key data */

2011-12-19T17:27:48Z

Key data

@@ Line 60: / Line 60: @@
 |-
 | higher antiderivatives || no antiderivatives higher than the first are expressible in terms of [[elementary function]]s.
 |}

Vipul: /* Definition in terms of the unit circle */

2011-12-19T17:13:47Z

Definition in terms of the unit circle

← Older revision		Revision as of 17:13, 19 December 2011
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	<math>\! \tan x := \frac{\sin x}{\cos x}</math>		<math>\! \tan x := \frac{\sin x}{\cos x}</math>

	===~~Definition in terms of the unit~~ circle===		===Unit circle definition===

	The '''~~sine~~ function''', denoted <math>\~~sin~~</math>, is defined as follows.		The '''tangent function''', denoted <math>\tan</math>, is defined as follows.

	Consider the unit circle centered at the origin, described as the following subset of the coordinate:		Consider the unit circle centered at the origin, described as the following subset of the coordinate:
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	<math>\{ (x,y) \mid x^2 + y^2 = 1\}</math>		<math>\{ (x,y) \mid x^2 + y^2 = 1\}</math>

	For a real number <math>t</math>, we define <math>\~~sin~~ t</math> as follows:		For a real number <math>t</math>, we define <math>\tan t</math> as follows:

	* Start at the point <math>(1,0)</math>, which lies on the unit circle centered at the origin.		* Start at the point <math>(1,0)</math>, which lies on the unit circle centered at the origin.