text: # When is tan undefined?
  
  The function y = tan(x) is undefined at all points where cos(x) = 0. This is because the tangent of an angle is defined as the sine of that angle divided by the cosine of that angle. In other words, tan(x) = sin(x)/cos(x).
  
  The value of a trigonometric function is undefined when the ratio for the function involves division by zero. The function thus has an asymptote at such a point.
  
  The trigonometric function y = tan(x) is undefined at all points where cos(x) = 0. This is because tan(x) is also defined as sin(x) divided by cos(x); namely, tan(x) = \frac{sin(x)}{cos(x)} .
  
  To determine the points where tan(x) is undefined, we solve for the equation cos(x) = 0. This gives us x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2},\pm\frac{5\pi}{2}, \ldots  . Therefore, the function y=tan(x) is undefined at all points x = \frac{\pi}{2} + k\pi , where k is an integer.
  
  Other properties of the tangent function are as follows:
  
  1) Its domain consists of all real numbers, except the zeroes of the cosine function: that is, all real numbers except \frac{\pi}{2} + k\pi , where k is an integer. Meanwhile, its range consists of all real numbers.
  
  2) The interval between two matching points on its graph (period) is \pi.
  
  3) Behavior of the graph: from the points 0 to \frac{\pi}{2} , the graph increases from 0 to positive infinity; from the points \frac{\pi}{2} to \pi , the graph increases from negative infinity to 0; and from \pi to \frac{3\pi}{2} , the graph increases from 0 to positive infinity. Therefore, from the points \frac{\pi}{2} to \frac{3\pi}{2} , the graph increases from negative infinity to positive infinity. In other words, the tangent is an increasing function between any two successive vertical asymptotes (please refer to the attached graph).
  
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distilabel_metadata.content: # When is tan undefined?
  
  The function y = tan(x) is undefined at all points where cos(x) = 0. This is because the tangent of an angle is defined as the sine of that angle divided by the cosine of that angle. In other words, tan(x) = sin(x)/cos(x).
  
  The value of a trigonometric function is undefined when the ratio for the function involves division by zero. The function thus has an asymptote at such a point.
  
  The trigonometric function y = tan(x) is undefined at all points where cos(x) = 0. This is because tan(x) is also defined as sin(x) divided by cos(x); namely, tan(x) = \frac{sin(x)}{cos(x)} .
  
  To determine the points where tan(x) is undefined, we solve for the equation cos(x) = 0. This gives us x = \pm\frac{\pi}{2}, \pm\frac{3\pi}{2},\pm\frac{5\pi}{2}, \ldots  . Therefore, the function y=tan(x) is undefined at all points x = \frac{\pi}{2} + k\pi , where k is an integer.
  
  Other properties of the tangent function are as follows:
  
  1) Its domain consists of all real numbers, except the zeroes of the cosine function: that is, all real numbers except \frac{\pi}{2} + k\pi , where k is an integer. Meanwhile, its range consists of all real numbers.
  
  2) The interval between two matching points on its graph (period) is \pi.
  
  3) Behavior of the graph: from the points 0 to \frac{\pi}{2} , the graph increases from 0 to positive infinity; from the points \frac{\pi}{2} to \pi , the graph increases from negative infinity to 0; and from \pi to \frac{3\pi}{2} , the graph increases from 0 to positive infinity. Therefore, from the points \frac{\pi}{2} to \frac{3\pi}{2} , the graph increases from negative infinity to positive infinity. In other words, the tangent is an increasing function between any two successive vertical asymptotes (please refer to the attached graph).
  
  Images:
  This image has been Flagged as inappropriate Click to unflag
  Image (1 of 1)
  Approved by eNotes Editorial Team
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distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed explanation of when the tangent function is undefined. It includes the mathematical reasoning behind the undefined points and lists additional properties of the tangent function. The document contains a question at the beginning and provides a comprehensive answer to it.
  
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