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The graph of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856401.png" /> (see Fig.). The sinusoid is a continuous curve with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856402.png" />. It intersects the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856403.png" />-axis at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856404.png" />. These are also points of inflection, meeting the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856405.png" />-axis at an angle of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856406.png" />. The extrema are at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856407.png" />.
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{{TEX|done}}
  
<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/s085640a.gif" />
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The graph of the function $y=\sin x$ (see Fig.). The sinusoid is a continuous curve with period $T=2\pi$. It intersects the $x$-axis at the points $(k\pi, 0)$. These are also points of inflection, meeting the $x$-axis at an angle of $\pm\pi/4$. The extrema are at the points $((k+1/2)\pi, (-1)^k)$.
  
Figure: s085640a
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<asy>
 +
import graph;
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size(500,200,IgnoreAspect);
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real f(real x) {return sin(x);};
  
The graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856408.png" /> is the cosinusoid, obtained by shifting the sinusoid a distance of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s0856409.png" /> to the left. The cosinusoid intersects the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564010.png" />-axis at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564011.png" />, and its extrema are at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564012.png" />.
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real f1(real x) {return cos(x);};
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draw(graph(f1,-2*pi,2*pi),blue,"$\cos(x)$");
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draw(graph(f,-2*pi,2*pi),red,"$\sin(x)$");
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xaxis("$x$",Arrow);
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yaxis();
  
Many oscillatory processes can be described by a periodic function of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564016.png" /> are constants and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564017.png" />. The graph of this function (called a general sinusoid) is obtained from the graph of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564018.png" /> (the ordinary sinusoid) as follows: expand in the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564019.png" />-axis by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564020.png" />, contract in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564021.png" />-direction by a factor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564022.png" />, translate to the left over a distance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564023.png" />, and when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564024.png" />: reflect in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564025.png" />-axis. Its period is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564026.png" /> and it meets the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564027.png" />-axis at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564028.png" />. Its extrema are at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564029.png" />.
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xtick("$\frac{\pi}{6}$",pi/6,N);
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xtick("$\frac{\pi}{4}$",pi/4,N);
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xtick("$\frac{\pi}{3}$",pi/3,N);
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xtick("$\frac{\pi}{2}$",pi/2,N);
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xtick("$\frac{3\pi}{2}$",3*pi/2,N);
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xtick("$\pi$",pi,N);
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xtick("$2\pi$",2*pi,N);
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xtick("$-\frac{\pi}{2}$",-pi/2,N);
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xtick("$-\frac{3\pi}{2}$",-3*pi/2,N);
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xtick("$-\pi$",-pi,N);
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xtick("$-2\pi$",-2*pi,N);
  
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ytick("$1/2$",0.5,1);
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ytick("$\sqrt{2}/2$",sqrt(2)/2,1);
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ytick("$\sqrt{3}/2$",sqrt(3)/2,1);
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ytick("$1$",1,1);
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ytick("$-1/2$",-0.5,-1);
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ytick("$-\sqrt{2}/2$",-sqrt(2)/2,-1);
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ytick("$-\sqrt{3}/2$",-sqrt(3)/2,-1);
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ytick("$-1$",-1,-1);
  
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attach(legend(),truepoint(E),10E,UnFill);
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</asy>
  
====Comments====
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The graph of $y=\cos x = \sin(x+\pi/2)$ is the cosinusoid, obtained by shifting the sinusoid a distance of $\pi/2$ to the left. The cosinusoid intersects the $x$-axis at the points $((k+1/2)\pi,0)$, and its extrema are at the points $(k\pi, (-1)^k)$.
In this article, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085640/s08564030.png" />.
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Many oscillatory processes can be described by a periodic function of the form y=a\sin(bx+c), where $a$, $b$ and $c$ are constants and $b>0$. The graph of this function (called a general sinusoid) is obtained from the graph of $y=\sin x$ (the ordinary sinusoid) as follows: expand in the direction of the $y$-axis by a factor $|a|$, contract in the $x$-direction by a factor $b$, translate to the left over a distance $c/b$, and when $a<0$: reflect in the $x$-axis. Its period is $T=2\pi/b$ and it meets the $x$-axis at the points $((k\pi-c)/b,0)$. Its extrema are at the points $(((k+1/2)\pi-c)/b,(-1)^ka)$.
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In this article, $k\in \mathbb Z$.
  
 
See also [[Sine|Sine]]; [[Trigonometric functions|Trigonometric functions]].
 
See also [[Sine|Sine]]; [[Trigonometric functions|Trigonometric functions]].

Revision as of 18:49, 5 December 2012


The graph of the function $y=\sin x$ (see Fig.). The sinusoid is a continuous curve with period $T=2\pi$. It intersects the $x$-axis at the points $(k\pi, 0)$. These are also points of inflection, meeting the $x$-axis at an angle of $\pm\pi/4$. The extrema are at the points $((k+1/2)\pi, (-1)^k)$.

The graph of $y=\cos x = \sin(x+\pi/2)$ is the cosinusoid, obtained by shifting the sinusoid a distance of $\pi/2$ to the left. The cosinusoid intersects the $x$-axis at the points $((k+1/2)\pi,0)$, and its extrema are at the points $(k\pi, (-1)^k)$.

Many oscillatory processes can be described by a periodic function of the form y=a\sin(bx+c), where $a$, $b$ and $c$ are constants and $b>0$. The graph of this function (called a general sinusoid) is obtained from the graph of $y=\sin x$ (the ordinary sinusoid) as follows: expand in the direction of the $y$-axis by a factor $|a|$, contract in the $x$-direction by a factor $b$, translate to the left over a distance $c/b$, and when $a<0$: reflect in the $x$-axis. Its period is $T=2\pi/b$ and it meets the $x$-axis at the points $((k\pi-c)/b,0)$. Its extrema are at the points $(((k+1/2)\pi-c)/b,(-1)^ka)$.

In this article, $k\in \mathbb Z$.

See also Sine; Trigonometric functions.

How to Cite This Entry:
Sinusoid. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sinusoid&oldid=15670
This article was adapted from an original article by Yu.A. Gor'kov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article