Difference between revisions of "Graphs of the Tangent, Cotangent, Cosecant and Secant Functions"
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+ | === Cotangent === | ||
+ | [[File:Graph of tangent.png|thumb|Graph of tangent]] | ||
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+ | The '''tangent''' of an angle is equivalent to the sine of that angle divided by the cosine of that angle. | ||
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+ | <math>\tan x = \frac{\sin{x}}{\cos{x}}</math> | ||
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+ | The range of the function <math>y = \tan{x}</math> is <math>(-\infty ,\infty)</math>. Note that since <math>\tan{x} = \frac{\sin{x}}{\cos{x}}</math>, <math>y = \tan{x}</math> is not defined whenever <math>\cos{x} = 0</math>; that is, when <math> x = \pi k + \frac{\pi}{2} </math> for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of <math>y = \tan{x}</math>. | ||
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+ | === Cotangent === | ||
+ | [[File:Graph of cotangent.png|thumb|Graph of cotangent]] | ||
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+ | The '''cotangent''' of an angle is the reciprocal of its tangent. | ||
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+ | <math>\cot x = \frac{1}{\tan x}</math> | ||
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+ | The range of the function <math>y = \cot{x}</math> is <math>(-\infty ,\infty)</math>, the same as the range of its reciprocal <math>\tan{x}</math>. <math>\cot{x} = \frac{1}{\tan{x}} = \frac{\cos{x}}{\sin{x}}</math>, so <math>y = \cot{x}</math> is not defined whenever <math>\sin{x} = 0</math>; that is, when <math> x = \pi k </math> for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of <math>y = \cot{x}</math>. | ||
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=== Secant === | === Secant === | ||
[[File:Graph of secant.png|thumb|Graph of secant]] | [[File:Graph of secant.png|thumb|Graph of secant]] | ||
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==Resources== | ==Resources== | ||
* [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Graphs%20of%20the%20Tangent,%20Cotangent,%20Cosecant%20and%20Secant%20Functions/Esparza%201093%20Notes%202.5.pdf Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]. Written notes created by Professor Esparza, UTSA. | * [https://mathresearch.utsa.edu/wikiFiles/MAT1093/Graphs%20of%20the%20Tangent,%20Cotangent,%20Cosecant%20and%20Secant%20Functions/Esparza%201093%20Notes%202.5.pdf Graphs of the Tangent, Cotangent, Cosecant and Secant Functions]. Written notes created by Professor Esparza, UTSA. | ||
+ | * [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Trigonometry |
Revision as of 16:32, 5 October 2021
Contents
Cotangent
The tangent of an angle is equivalent to the sine of that angle divided by the cosine of that angle.
The range of the function is . Note that since , is not defined whenever ; that is, when for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of .
Cotangent
The cotangent of an angle is the reciprocal of its tangent.
The range of the function is , the same as the range of its reciprocal . , so is not defined whenever ; that is, when for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of .
Secant
The secant of an angle is the reciprocal of its cosine.
The range of the function is , so the range of is , which is all possible outputs of 1/x for . Since is not defined, is not defined whenever ; that is, when for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of .
Cosecant
The cosecant of an angle is the reciprocal of its sine.
The range of the function is the same as the range for (). is not defined whenever ; that is, when for any integer k. We can see that there is a vertical asymptote at each of these values in the graph of .
Resources
- Graphs of the Tangent, Cotangent, Cosecant and Secant Functions. Written notes created by Professor Esparza, UTSA.
- [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Trigonometry