Difference between revisions of "Graphs of the Sine and Cosine Functions"

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[[File:Horizontal stretch of sinx.png|thumb|<math> \sin{x} </math> (red) and <math> \sin{\left(\frac{1}{2}x\right)} </math> (green, horizontal stretch of sin(x) by factor of 2). Horizontal stretches/compressions of sine and cosine functions change the period.]]
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[[File:Vertical stretch of sinx.png|thumb|<math> \sin{x} </math> (red) and <math> \sin{\left(2x\right)} </math> (green, vertical stretch of sin(x) by factor of 2). Vertical stretches/compressions of sine and cosine functions change the amplitude.]]
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The sine and cosine functions have several distinct characteristics:
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* They are periodic functions.
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* The domain of each function is <math> (-\infty, \infty) </math> and the range is [−1,1].
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* The graph of <math> y = \sin{x} </math> is symmetric about the origin.
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* The graph of <math> y = \cos{x} </math> is symmetric about the y-axis.
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* Given <math> y = A\sin{(\omega x)} </math> or <math> y = A\cos{(\omega x)} </math> for some real numbers <math> A </math> and <math>\omega</math>:
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:* The '''amplitude''' of the function is <math> |A| </math>, and represents the maximum distance from the x-axis that the function covers.
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:* The '''period''' (the distance between two peaks or two valleys) is <math> \frac{2\pi}{\omega} </math>, and represents one full "cycle" of the function.
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The graph of the sine function looks like this:
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:[[File:sin.svg|Graph of the sine function|500px]]
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Careful analysis of this graph will show that the graph corresponds to the unit circle. x is essentially the degree measure (in radians), while y is the value of the sine function.
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The graph of the cosine function looks like this:
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:[[File:cos.svg|Graph of the cosine function|500px]]
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As with the sine function, analysis of the cosine function will show that the graph corresponds to the unit circle. One of the most important differences between the sine and cosine functions is that sine is an odd function (i.e. <math>\sin(-\theta)=-\sin(\theta)</math> while cosine is an even function (i.e. <math>\cos(-\theta)=\cos(\theta)</math> .
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Sine and cosine are periodic functions; that is, the above is repeated for preceding and following intervals with length <math>2\pi</math> .
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:[[File:Sine_cosine_plot.svg|Sine and cosine on same graph|500px]]
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==Resources==
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Graphs%20of%20the%20Sine%20and%20Cosine%20Functions/Esparza%201093%20Notes%202.4.pdf Graphs of the Sine and Cosine]. Written notes created by Professor Esparza, UTSA.
 
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Graphs%20of%20the%20Sine%20and%20Cosine%20Functions/Esparza%201093%20Notes%202.4.pdf Graphs of the Sine and Cosine]. Written notes created by Professor Esparza, UTSA.
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* [https://courses.lumenlearning.com/precalctwo/chapter/graphs-of-the-sine-and-cosine-function/ Graphs of the Sine and Cosine Functions], Lumen Learning
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==Licensing==
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Content obtained and/or adapted from:
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* [https://courses.lumenlearning.com/precalctwo/chapter/graphs-of-the-sine-and-cosine-function/ Graphs of the Sine and Cosine Functions, Lumen Learning Precalculus II] under a CC BY license
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* [https://en.wikibooks.org/wiki/Trigonometry/Graphs_of_Sine_and_Cosine_Functions Graphs of sine and cosine functions, Wikibooks: Trigonometry] under a CC BY-SA license

Latest revision as of 13:07, 25 October 2021

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{x} } (red) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\left(\frac{1}{2}x\right)} } (green, horizontal stretch of sin(x) by factor of 2). Horizontal stretches/compressions of sine and cosine functions change the period.
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{x} } (red) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin{\left(2x\right)} } (green, vertical stretch of sin(x) by factor of 2). Vertical stretches/compressions of sine and cosine functions change the amplitude.

The sine and cosine functions have several distinct characteristics:

  • They are periodic functions.
  • The domain of each function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-\infty, \infty) } and the range is [−1,1].
  • The graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \sin{x} } is symmetric about the origin.
  • The graph of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = \cos{x} } is symmetric about the y-axis.
  • Given Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = A\sin{(\omega x)} } or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y = A\cos{(\omega x)} } for some real numbers Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A } and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega} :
  • The amplitude of the function is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |A| } , and represents the maximum distance from the x-axis that the function covers.
  • The period (the distance between two peaks or two valleys) is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{2\pi}{\omega} } , and represents one full "cycle" of the function.

The graph of the sine function looks like this:

Graph of the sine function

Careful analysis of this graph will show that the graph corresponds to the unit circle. x is essentially the degree measure (in radians), while y is the value of the sine function.

The graph of the cosine function looks like this:

Graph of the cosine function

As with the sine function, analysis of the cosine function will show that the graph corresponds to the unit circle. One of the most important differences between the sine and cosine functions is that sine is an odd function (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sin(-\theta)=-\sin(\theta)} while cosine is an even function (i.e. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \cos(-\theta)=\cos(\theta)} .

Sine and cosine are periodic functions; that is, the above is repeated for preceding and following intervals with length Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2\pi} .

Sine and cosine on same graph


Resources

Licensing

Content obtained and/or adapted from:

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