View source for Graphs of the Tangent, Cotangent, Cosecant and Secant Functions

=== Tangent ===
[[File:Graph of tangent.png|thumb|Graph of tangent]]

The '''tangent''' of an angle is equivalent to the sine of that angle divided by the cosine of that angle.

<math>\tan x = \frac{\sin{x}}{\cos{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>.




=== Cotangent ===
[[File:Graph of cotangent.png|thumb|Graph of cotangent]]

The '''cotangent''' of an angle is the reciprocal of its tangent.

<math>\cot x = \frac{1}{\tan 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>.




=== Secant ===
[[File:Graph of secant.png|thumb|Graph of secant]]

The '''secant''' of an angle is the reciprocal of its cosine.

<math>\sec x = \frac{1}{\cos x}</math>

Like <math> y=\cos{x} </math>, <math> y = \sec{x} </math> is an even function; that is, <math> \sec{(x)} = \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>.



=== Cosecant ===
[[File:Graph of cosecant.png|thumb|Graph of cosecant]]

The '''cosecant''' of an angle is the reciprocal of its sine.

<math>\csc{x} = \frac{1}{\sin{x}}</math>

Like <math> y=\sin{x} </math>, <math> y = \csc{x} </math> is an odd function; that is, <math> \csc{(x)} = -\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>.



==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://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Trigonometry Trigonometry], WikiBooks: Pure Mathematics 2


==Licensing==
Content obtained and/or adapted from:
* [https://en.wikibooks.org/wiki/A-level_Mathematics/CIE/Pure_Mathematics_2/Trigonometry Trigonometry, WikiBooks: Pure Mathematics 2] under a CC BY-SA license