Graphs of Tangent and Cotangent Functions
Definitions of Tangent and Cotangent
Recall that the tangent and cotangent functions are defined in terms of the sine and cosine:
where
We see from the definitions that cotangent is the reciprocal of tangent, that is
Let us consider the properties of these two functions in more detail.
The Graph and Properties of the Tangent Function
Domain and Codomain of the Tangent Function
The tangent function is not defined at the points
where dom denotes the domain and codom denotes the codomain or range of the function.
Monotonicity of the Tangent Function
Show that the tangent function is increasing in the open interval
The angles
We see that if
which means that the tangent function is strictly increasing in the given interval.
Parity of the Tangent Function
It is easy to prove that the tangent function is odd:
As any other odd function, the graph of tangent function is symmetric about the origin.
Periodicity of the Tangent Function
The tangent function is periodic with the least period
where
Zeros of the Tangent Function
The tangent will be zero wherever its numerator (the sine) is zero. Hence, the roots of the equation
Graph of the Tangent Function
Since
The Graph and Properties of the Cotangent Function
Domain and Codomain of the Cotangent Function
The function
Monotonicity of the Cotangent Function
The cotangent is a decreasing function between any two adjacent points of discontinuity. We prove this as follows. Let points
The angles
Thus, the fraction in the right-hand side of the above formula is positive. Given a minus sign before the fraction, we conclude that
Hence, the cotangent function is strictly decreasing in the interval
Parity of the Cotangent Function
The function
The graph of cotangent function is symmetric about the origin.
Periodicity of the Cotangent Function
Similarly to tangent, the cotangent function is periodic with the least period
where
Zeros of the Cotangent Function
The cotangent will have zeros at the points where cosine is zero. Therefore, the solution of the equation
Graph of the Cotangent Function
The cotangent graph has vertical asymptotes at the points