## 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.
\( \cot(\theta)\) 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 \( \theta = -0.1 \)
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.
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 \( y = \cot(x) \). ## Properties of cot x
1) cot x has a period equal to \( \pi \).
## 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. ## More References and Links Related to the Cotangent cot x functionSine Function sin xCosine Function cos x Trigonometric Functions. |