Is the tangent function considered even, odd, or neither?

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The tangent function, denoted as tan(x), is classified as an odd function. This classification is based on its mathematical properties and the symmetry of its graph.

To understand this, we can consider the definition of an odd function. A function f(x) is called odd if it satisfies the condition:

f(-x) = -f(x)

For the tangent function, we can apply this definition:

tan(-x) = -tan(x)

This equality shows that if we take the tangent of the negative angle -x, it is equal to the negative of the tangent of the angle x. Hence, since the condition for being odd is satisfied, we conclude that:

Key Characteristics of the Tangent Function:

  • Graph: The graph of tan(x) is symmetric with respect to the origin, which is a typical attribute of odd functions.
  • Periodicity: The tangent function is periodic with a period of π. This means repeating values in intervals of π.
  • Range: The range of the tangent function is all real numbers, with vertical asymptotes at (2n+1)π/2, where n is an integer.

To further illustrate, consider some values:

  • tan(0) = 0
  • tan(π/4) = 1 and tan(-π/4) = -1
  • tan(π/3) = √3 and tan(-π/3) = -√3

Each of these examples reinforces that for every positive input, there is a corresponding negative input that results in a negative output, thus confirming that the tangent function is indeed an odd function.

In summary, tan(x) is an odd function, meaning it exhibits rotational symmetry around the origin and satisfies the property:

tan(-x) = -tan(x)

This makes it distinct from even functions, which exhibit symmetry about the y-axis.

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