Tangent function in trigonometry
Tangent function is one of the basic trigonometric function along with sine and cosine function. For a circle of radius 1, tangent equals the ratio of y and x coordinates. So it is also known as circular function.
From above diagram,
tanθ = length of opposite side / length of adjacent side
Details on period of tangent function
Tangent function repeats after every π interval, so period of tangent function is π. This will more clear when we see process of graphing a tangent function. Cotangent is inverse function of tangent function. So tanθ = 1/cotθ and also tanθ = cot(90°θ).
Example : solve the value of tan60° using cot function analogy.
Now we learn another type of tangent function known as hyperbolic tangent function. It is derived from two basic hyperbolic functions, sinh and cosh. tanh is a ratio of sinh and cosh.
tanhθ = sinhθ/coshθ
in above equation hyperbolic sine function sinhθ = eθ eθ2.
And hyperbolic cosine function coshθ = eθ+ eθ2.
The process of graphing tangent function includes following steps.

Find the values of tangent for some standard value of angles.

Plot the graph of angles verses values of tangent function.
From the process of graphing tangent functions, we can plot graph as shown below.
By the process of graphing tan functions, we can observe some properties of tangent function very easily.
1) Tangent function repeats after every π radian interval, so period of a tangent function is π radians.
2) Tangent function exists for all real values except at π/2 + k π, where k is integer vales from ∞ to +∞. So, domain of tangent function is called (π/2, π/2).
3) Tangent function has a values from ∞ to +∞, so range of tangent function is called (∞,+∞).
4) Tangent function is odd function so tan(θ) = tan(θ).
5) Tangent function is symmetric around origin (0,0).
6) Tangent function is undefined at π/2 + k π. So it is having vertical asymptotes at x =  π/2 + k π.
Tangent Graph video lesson
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