Vipul: /* Key data */

2012-02-12T22:41:55Z

Key data

← Older revision		Revision as of 22:41, 12 February 2012
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	\| [[period]] \|\| <math>\pi</math>		\| [[period]] \|\| <math>\pi</math>
	\|-		\|-
	\| important symmetries \|\| [[even function]] (see [[composite of even function and odd function is even]]<br>The function has mirror symmetry with respect to vertical lines for <math>x</math>-values at ''all'' multiples of <math>\pi/2</math>, including the even and odd multiples.		\| important symmetries \|\| [[even function]] (see [[composite of even function and odd function is even]])<br>The function has mirror symmetry with respect to vertical lines for <math>x</math>-values at ''all'' multiples of <math>\pi/2</math>, including the even and odd multiples.
	\|-		\|-
	\| [[local maximum value]]s and points of attainment \|\| local maximum value of 1, attained at all odd integer multiples of <math>\pi/2</math>		\| [[local maximum value]]s and points of attainment \|\| local maximum value of 1, attained at all odd integer multiples of <math>\pi/2</math>

Vipul: Created page with "{{particular function}} ==Definition== This function of one variable, denoted $|\sin|$ , is defined as the [[defining ingredient::composite of two functions|compo..."

2011-09-22T17:16:31Z

Created page with "{{particular function}} ==Definition== This function of one variable, denoted <math>|\sin|</math>, is defined as the [[defining ingredient::composite of two functions|compo..."

New page

{{particular function}}

==Definition==

This [[function]] of one variable, denoted <math>|\sin|</math>, is defined as the [[defining ingredient::composite of two functions|composite]] of the [[defining ingredient::absolute value function]] and the [[defining ingredient::sine function]], i.e., it is defined as:

<math>x \mapsto |\sin x|</math>

The function can also be defined using a [[piecewise definition of function|piecewise definition]] as follows:

<math>|\sin x| = \left\lbrace \begin{array}{rl}\sin x, & 2n\pi \le x \le (2n + 1)\pi, n \in \mathbb{Z}\\ -\sin x, & (2n - 1)\pi < x < 2n\pi, n \in \mathbb{Z}\\\end{array}\right.</math>

==Key data==

{| class="sortable" border="1"
! Item !! Value
|-
| default [[domain]] || all [[real number]]s, i.e., all of <math>\R</math>
|-
| [[range]] || the [[closed interval]] <math>[0,1]</math>, i.e., <math>\{ y : 0 \le y \le 1 \}</math><br>[[absolute maximum value]]: 1, [[absolute minimum value]]: 0
|-
| [[period]] || <math>\pi</math>
|-
| important symmetries || [[even function]] (see [[composite of even function and odd function is even]]<br>The function has mirror symmetry with respect to vertical lines for <math>x</math>-values at ''all'' multiples of <math>\pi/2</math>, including the even and odd multiples.
|-
| [[local maximum value]]s and points of attainment || local maximum value of 1, attained at all odd integer multiples of <math>\pi/2</math>
|-
| [[local minimum value]]s and points of attainment || local minimum value of 0, attained at all integer multiples of <math>\pi</math> (or equivalently, all ''even'' integer multiples of <math>\pi/2</math>)
|-
| [[first derivative]] || <math>x \mapsto \left\lbrace \begin{array}{rl}\cos x, & 2n\pi < x < (2n + 1)\pi, n \in \mathbb{Z}\\ -\cos x, & (2n - 1)\pi < x < 2n\pi, n \in \mathbb{Z}\\\end{array}\right.</math> The function is not differentiable at multiples of <math>\pi</math>. At any multiple of <math>\pi</math>, the [[left hand derivative]] is -1 and the [[right hand derivative]] is +1.
|-
| [[second derivative]] || <math>x \mapsto -|\sin x|</math> except at integer multiples of <math>\pi</math>, where the second derivative is undefined.
|}

Absolute value of sine function - Revision history

Vipul: /* Key data */

Vipul: Created page with "{{particular function}} ==Definition== This function of one variable, denoted |\sin|, is defined as the [[defining ingredient::composite of two functions|compo..."

Vipul: Created page with "{{particular function}} ==Definition== This function of one variable, denoted $|\sin|$ , is defined as the [[defining ingredient::composite of two functions|compo..."