Cotangent Function cot x
Definition and Graph of the Cotangent Function
Angle \( \theta \) with initial side on the positive x axis (in standard position) and terminal side OM is shown below.
Given point \( M(x,y) \) on the terminal side, the cotangent function is defined as
\[ \cot(\theta) = \dfrac{x}{y} \]
\( \cot(\theta) \) may also be expressed in terms of \( \sin(\theta) \) and \( \cos(\theta) \) as follows:
\[ \cot(\theta) = \dfrac{x}{y} = \dfrac{x/r}{y/r} = \dfrac{\cos(\theta)}{\sin(\theta)}\]
Note that
1) \[ \cot(\theta+\pi) = \dfrac{\cos(\theta+\pi)}{\sin(\theta+\pi)} = \dfrac{-\cos(\theta)}{-\sin(\theta)}= \dfrac{\cos(\theta)}{\sin(\theta)} = \cot(\theta)\]
and we therefore conclude that \( \cot(\theta) \) is a periodic function with a period equal to \( \pi \).
2) \[ \cot(-\theta) = \dfrac{\cos(-\theta)}{\sin(-\theta)} = \dfrac{ \cos(\theta)}{-\sin(\theta)} = - \dfrac{\cos(\theta)}{\sin(\theta)} = - \cot(\theta)\]
and therefore \( \cot(\theta) \) is an odd function which also means that the graph of \( \cot(\theta) \) is symmetric with respect to the origin of the system of coordinates.
We now use a unit circle to find \( \sin(\theta)\) and \( \cos(\theta)\) and hence \( \cot(\theta)\) over one period extending from \( \theta = 0 \) to \( \theta = \pi \).
We know from the sine and cosine functions that the x and y coordinates on a unit circle gives the values of \( \sin(\theta)\) and \( \cos(\theta)\) as shown below.
Let us now put the values of the angles \( 0, \dfrac{\pi}{4} , \dfrac{\pi}{2} , \dfrac{3\pi}{4} , \pi \) and the corresponding values of \( \cos(\theta) \) and \( \sin(\theta) \) on a table as shown below.
| \( \theta \) | \( \cos(\theta) \) | \( \sin(\theta) \) | \( \cot(\theta) = \dfrac{\cos(\theta)}{\sin(\theta)}\) |
|
| \( 0 \) | \( 1 \) | \( 0 \) | \( undefined \) |
| \( \dfrac{\pi}{4} \) | \( \dfrac{\sqrt 2}{2} \) | \( \dfrac{\sqrt 2}{2} \) | \( 1 \) |
| \( \dfrac{\pi}{2} \) | \( 0 \) | \( 1 \) | \( 0 \) |
| \( \dfrac{3\pi}{4} \) | \( - \dfrac{\sqrt 2}{2} \) | \( \dfrac{\sqrt 2}{2} \) | \( - 1 \) |
| \( \pi \) | \( - 1 \) | 7 \( 0 \) | \( undefined \) |
\( \cot(\theta)\) is undefined at \( \theta = 0 \) and \( \theta = \pi \), however we can get information about the behaviour \( \cot(\theta)\) close to these values.
We use the calculator to find values of \( \cot(\theta)\) as \( \theta \) approaches \( 0 \) starting at \( \theta = -0.1 \)
| \( \theta \) | \( \cot(\theta) \) |
|
| \( -0.1 \) | \( -9.966644423 \) |
| \( -0.01\) | \( -99.99666664 \) |
| \( -0.001 \) | \( -999.9996667 \) |
| \( -0.000001 \) | \( -1000000 \) |
As \( \theta \) approaches \( 0 \) by values smaller than \( 0 \), \( \cot(\theta) \) approaches small values.
Let us now consider values that approaches 0 such that these values are larger than 0.
| \( \theta \) | \( \cot(\theta) \) |
|
| \( 0.1 \) | \( 9.966644423 \) |
| \( 0.01\) | \( 99.99666664 \) |
| \( 0.001 \) | \( 999.9996667 \) |
| \( 0.000001 \) | \( 1000000 \) |
As \( \theta \) approaches \(0 \) by values larger than \( 0 \) , \( \cot(\theta) \) approaches large values and hence the existence of a vertical asymptote at \( \theta = 0 \).
Using the concept of limits, we describe the behaviour of \( \cot(\theta) \) as \( \theta \) approaches \( 0 \) from the left (or by values smaller than \( 0 \)) as follows
\( \lim_{\theta \to 0^-} \cot(\theta) = -\infty \)
and the behaviour of \( \cot(\theta) \) as \( \theta \) approaches \( 0 \) from the right (or by values larger than \( 0 \)) as follows
\( \lim_{\theta \to 0^+} \cot(\theta) = +\infty \)
We now use a system of rectangular axes \( (x,y) \) to plot the points in the above table and approximate the graph of the cotangent cot x function as shown below.
NOTE
We are used to \( x \) being the variable of a function, \(x\) on the graph takes values of \( \theta \) and y takes the values of \( \cot(\theta) \) which is labeled as \( y = \cot(x) \).
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Properties of cot x
1) cot x has a period equal to \( \pi \).
2) \( \cot(x) \) has vertical asymptotes at all values of \( x = n\pi \) , \( n \) being any integer.
3) The domain of \( \cot(x) \) is the set of all real numbers except \( x = n\pi \) , \( n \) being any integer.
4) The graph of \( \cot(x) \) is symmetric with respect to the origin of the system of coordinates.
5) The range of \( \cot(x) \) is given by: \( (-\infty , +\infty) \)
6) \( \cot(x) \) is odd and its graph is symmetric with respect to the origin of the system of axes.
7) \( \cot(x) \) is decreasing on intervals.
Interactive Tutorial on the General Cotangent Function
The general cotangent function given by
\( f ( x ) = a \cot ( b x + c ) + d \)
and its period, phase shift, asymptotes domain and range are explored interactively. An app is used, where parameters a, b, c and d are changed to investigate their effects on the graph of f.
1 - Set a = 1, b = 1, c = 0 and d = 0. Take note of the period, phase shift and positions of the asymptotes (vertical lines) of the graph of f? Now change a , how does it affect the graph? Does it affect its range? If yes, how?
2 - Set a = 1, c = 0, d = 0 and change b. Approximate the period from the graph and compare it to \( \dfrac{2\pi}{| b |} \). How does b affect the graph of f(x)? How does it affect the asymptotes of the graph of f?
3 - Set a = 1, b = 1, d = 0 and change c starting from zero increasing slowly to positive large values. Take note of the shift, is it to the left or to the right? Compare its measure to \( - c / b \).
4 - Set a = 1, b = 1, d = 0 and change c starting from zero deceasing slowly to negative smaller values. Take note of the shift, is it a left or a right shift? Compare its measure to \( - c / b \).
5 - Set a,b and c to non zero values and change d. What is the direction of the shift of the graph?
6 - Which parameters affect the positions of the vertical asymptotes? Explain analytically.
7 - Which parameters affect the domain of the cotangent function? Explain analytically.
8 - Which parameters affect the range of the cotagent function? Explain analytically.