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Difference between revisions of "Sinusoid" - Encyclopedia of Mathematics
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Difference between revisions of "Sinusoid"

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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 $((k+1/2)\pi, (-1)^k)$.
 
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 $((k+1/2)\pi, (-1)^k)$.
  
<asy>
+
<center><asy>
 
import graph;
 
import graph;
size(500,200,IgnoreAspect);
+
size(450);
 
real f(real x) {return sin(x);};
 
real f(real x) {return sin(x);};
  
 
real f1(real x) {return cos(x);};
 
real f1(real x) {return cos(x);};
draw(graph(f1,-2*pi,2*pi),blue,"$\cos(x)$");
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draw(graph(f1,-2*pi,2*pi),blue+1,"$\cos(x)$");
draw(graph(f,-2*pi,2*pi),red,"$\sin(x)$");
+
draw(graph(f,-2*pi,2*pi),red+1,"$\sin(x)$");
 
xaxis("$x$",Arrow);
 
xaxis("$x$",Arrow);
 
yaxis();
 
yaxis();
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xtick("$-2\pi$",-2*pi,N);
 
xtick("$-2\pi$",-2*pi,N);
  
ytick("$1/2$",0.5,1);
+
ytick("$1/2$",0.5,1,fontsize(8pt));
ytick("$\sqrt{2}/2$",sqrt(2)/2,1);
+
ytick("$\sqrt{2}/2$",sqrt(2)/2,1,fontsize(8pt));
ytick("$\sqrt{3}/2$",sqrt(3)/2,1);
+
ytick("$\sqrt{3}/2$",sqrt(3)/2,1,fontsize(8pt));
ytick("$1$",1,1);
+
ytick("$1$",1,1,fontsize(8pt));
ytick("$-1/2$",-0.5,-1);
+
ytick("$-1/2$",-0.5,-1,fontsize(8pt));
ytick("$-\sqrt{2}/2$",-sqrt(2)/2,-1);
+
ytick("$-\sqrt{2}/2$",-sqrt(2)/2,-1,fontsize(8pt));
ytick("$-\sqrt{3}/2$",-sqrt(3)/2,-1);
+
ytick("$-\sqrt{3}/2$",-sqrt(3)/2,-1,fontsize(8pt));
ytick("$-1$",-1,-1);
+
ytick("$-1$",-1,-1,fontsize(8pt));
  
 
attach(legend(),truepoint(E),10E,UnFill);
 
attach(legend(),truepoint(E),10E,UnFill);
</asy>
+
</asy></center>
  
 
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 $(k\pi, (-1)^k)$.
 
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 $(k\pi, (-1)^k)$.

Revision as of 16:25, 5 December 2014


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=35358
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