Vipul at 13:29, 5 September 2011

2011-09-05T13:29:18Z

← Older revision		Revision as of 13:29, 5 September 2011
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	* <math>\alpha = A \cos \varphi</math>.		* <math>\alpha = A \cos \varphi</math>.
	* <math>\beta = A \sin \varphi</math>.		* <math>\beta = A \sin \varphi</math>.

			==Examples==

			{\| class="sortable" border="1"
			! Function !! How it's a sinusoidal function in the linear transform sense !! How it's a sinusoidal function in the linear combination sense
			\|-
			\| [[sine function]] \|\| <math>0 + 1\sin(1x + 0)</math>:<br><math>\mu = 0, A = 1, m = 1, \varphi = 0</math> \|\| <math>0 + 1\sin(1x) + 0\cos(1x)</math><br><math>\mu = 0, m = 1, \alpha = 1, \beta = 0</math>
			\|-
			\| [[cosine function]] \|\| <math>0 + 1\sin(1x + \pi/2)</math><br><math>\mu = 0, A = 1, m = 1, \varphi = \pi/2</math> \|\| <math>0 + 0\sin(1x) + 1\cos(1x)</math><br><math>\mu = 0, m = 1, \alpha = 0, \beta = 1</math>.
			\|-
			\| [[sine-squared function]] <math>\sin^2</math> \|\| <math>1/2 + (1/2)\sin(2x - \pi/2)</math><br><math>\mu = 1/2, A = 1/2, m = 2, \varphi = -\pi/2</math> \|\| <math>(1/2) + 0\sin(2x) + (-1/2)\cos(2x)</math><br><math>\mu = 1/2, m = 2, \alpha = 0, \beta = -1/2</math>
			\|-
			\| [[cosine-squared function]] <math>\cos^2</math> \|\| <math>1/2 + (1/2)\sin(2x + \pi/2)</math><br><math>\mu = 1/2, A = 1/2, m = 2, \varphi = \pi/2</math> \|\| <math>(1/2) + 0\sin(2x) + (1/2)\cos(2x)</math><br><math>\mu = 1/2, m = 2, \alpha = 0, \beta = 1/2</math>
			\|}

Vipul: Created page with "==Definition== ===As a linear transform of the sine function=== The term '''sinusoidal function''' refers to a function of the form $f \circ \sin \circ g$ where
2011-09-05T13:22:12Z

Created page with "==Definition== ===As a linear transform of the sine function=== The term '''sinusoidal function''' refers to a function of the form <math>f \circ \sin \circ g</math> where <mat..."

New page
==Definition==

===As a linear transform of the sine function===

The term '''sinusoidal function''' refers to a function of the form <math>f \circ \sin \circ g</math> where <math>f</math> and <math>g</math> are [[linear function]]s and <math>\sin</math> is the [[sine function]]. Specifically, it is a function of the form:

<math>x \mapsto \mu + A\sin(mx + \varphi), \qquad A > 0, m \ne 0</math>

Here:

* <math>\mu</math> is the mean value about which the function is oscillating, i.e., the graph of a function looks like a scaled sine function about the horizontal line <math>y = \mu</math>.
* <math>A</math> is the ''amplitude'' of oscillations, i.e., the function oscillates between a minimum value of <math>\mu - A</math> and a maximum value of <math>\mu + A</math>.
* <math>m</math> is the ''angular frequency'' parameter and controls the [[period]] of oscillations, which is given by <math>2\pi/m</math>.
* <math>\varphi</math> is a ''phase parameter'' that roughly describes the ''head start'' of the function relative to a function that starts at its mean value at <math>x = 0</math>.

===As a linear combination of sine and cosine functions===

The term '''sinusoidal function''' can be used for a function of the form:

<math>x \mapsto \mu + \alpha \sin(mx) + \beta \cos(mx), \qquad m \ne 0, \alpha^2 + \beta^2 > 0</math>

===Conversion between the two versions===

Here's how we convert the linear combination version to the linear transform version:

* <math>\mu,m</math> remain the same.
* Set <math>A = \sqrt{\alpha^2 + \beta^2}</math>.
* Set <math>\varphi</math> as an angle so that <math>\cos \varphi = \alpha/A</math> and <math>\sin \varphi = \beta/A</math>. <math>\varphi</math> is uniquely determined up to additive multiples of <math>2\pi</math>.

Here's how we convert the linear transform version to the linear combination version:

* <math>\mu, m</math> remain the same.
* <math>\alpha = A \cos \varphi</math>.
* <math>\beta = A \sin \varphi</math>.