Difference between revisions of "Graphs of the Tangent, Cotangent, Cosecant and Secant Functions"
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<math>\sec x = \frac{1}{\cos x}</math> | <math>\sec x = \frac{1}{\cos x}</math> | ||
| − | Since <math> \frac{1}{0} </math> is not defined, <math>\sec{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 = \sec{x}</math>. | + | The range of <math> \cos{x} </math> is <math> [-1, 1] </math>, so the range of <math> \sec{x} </math> is <math> (-\infty , -1] \cup [1, \infty ) </math>, which is all possible outputs of 1/x for <math> -1\leq x\geq 1 </math>. Since <math> \frac{1}{0} </math> is not defined, <math>\sec{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 = \sec{x}</math>. |
=== Cosecant === | === Cosecant === | ||
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<math>\operatorname{cosec} x = \frac{1}{\sin x}</math> <ref group="note">Some sources may use <math>\csc x</math>, but this notation is not endorsed by Cambridge</ref> | <math>\operatorname{cosec} x = \frac{1}{\sin x}</math> <ref group="note">Some sources may use <math>\csc x</math>, but this notation is not endorsed by Cambridge</ref> | ||
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| + | <math>\csc{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 = \csc{x}</math>. | ||
=== Cotangent === | === Cotangent === | ||
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<math>\cot x = \frac{1}{\tan x}</math> | <math>\cot x = \frac{1}{\tan x}</math> | ||
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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. | ||
Revision as of 16:14, 5 October 2021
Secant, Cosecant, and Cotangent
Secant
The secant of an angle is the reciprocal of its cosine.
The range of 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 .
![{\displaystyle [-1,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51e3b7f14a6f70e614728c583409a0b9a8b9de01)
![{\displaystyle (-\infty ,-1]\cup [1,\infty )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e3448e2de68557e598967fb8b1f8900260c4a64)
Cosecant
The cosecant of an angle is the reciprocal of its sine.
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.
Resources
- Graphs of the Tangent, Cotangent, Cosecant and Secant Functions. Written notes created by Professor Esparza, UTSA.
Cite error: <ref> tags exist for a group named "note", but no corresponding <references group="note"/> tag was found, or a closing </ref> is missing
