Difference between revisions of "Graphs of the Tangent, Cotangent, Cosecant and Secant Functions"
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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>. | 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 === | === Cotangent === | ||
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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>. | 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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The range of the function <math> y = \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\leq 1 </math>. Since <math> \frac{1}{0} </math> is not defined, <math>y = \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 the function <math> y = \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\leq 1 </math>. Since <math> \frac{1}{0} </math> is not defined, <math>y = \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>. | ||
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The range of the function <math> y = \csc{x}</math> is the same as the range for <math>\sec{x}</math> (<math> (-\infty , -1] \cup [1, \infty ) </math>). <math> y = \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>. | The range of the function <math> y = \csc{x}</math> is the same as the range for <math>\sec{x}</math> (<math> (-\infty , -1] \cup [1, \infty ) </math>). <math> y = \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>. | ||
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Revision as of 16:35, 5 October 2021
Contents
Tangent
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