## Definition and Graph of the Coseacnt Function
Angle \( \theta \) with initial side on the positive x axis and terminal side OM is shown below. It is an angle in standard position.
\( \csc(\theta)\) is undefined at \( \theta = 0 \), \( \theta = \pi \) and \( \theta = 2\pi \) , however we can get information about the behaviour \( \csc(\theta)\) close to these values using a claculator. We use the calculator to find values of \( \csc(\theta)\) as \( \theta \) approaches \( 0 \) starting at \( \theta = -0.1 \)
As \( \theta \) approaches \( 0 \) by values smaller than \( 0 \), \( \csc(\theta) \) approaches small values. We now use the calculator to find values of \( \csc(\theta)\) as \( \theta \) approaches \( 0 \) starting at \( \theta = 0.1 \)
Using the concept of limits, we describe the behaviour of \( \csc(\theta) \) as \( \theta \) approaches \( 0 \) from the left (or by values smaller than \( 0 \)) as follows \( \lim_{\theta \to 0^-} \csc(\theta) = -\infty \) and the behaviour of \( \csc(\theta) \) as \( \theta \) approaches \( 0 \) from the right (or by values larger than \( 0 \)) as follows \( \lim_{\theta \to 0^+} \csc(\theta) = +\infty \) Similar behaviour occur close to all values of \( x = n\pi \) where \( n \) is an integer. We now use a system of rectangular coordinates \( (x,y) \) to plot the points in the above table and approximate the graph of the cosecant function \( \csc x \) as shown below. \( \csc (\theta) \) from \( \theta = 0 \) to \( \theta = 2\pi \) is shown in red along with \( \sin (\theta) \) in green to understand the relationship, of the two functions over one period, graphically. The most important observation is: the vertical asymptotes (shown in broken lines) of \( \csc (\theta) \) occur at the position of the zeros of \( \sin (\theta) \). NOTE Because 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 \( \csc(\theta) \) which is noted as \( y = \csc(x) \). The broken vertical lines indicate the vertical asymptotes of \( \csc x \). ## Properties of csc x
1) csc x has a period equal to \( 2\pi \).
## Interactive Tutorial on the Cosecant Function csc x of the General FormA tutorial on exploring the cosecant function of the general form given by\( f(x) = a \csc( b x + c ) + d \) is presented. An interactive app is used where parameters a, b, c and d may be changed and their effects on the period, phase shift, asymptotes, domain and range explored. Period is given by: \( \dfrac{2\pi}{|b|} \) Use the app to answer the questions below.
1 - Use the scroll bar to set a = 1, b = 1, c = 0 and d = 0. Now change a , how does it affect the graph? Does it affect the ## More references on the trigonometric functionsProperties of Trigonometric Functions.Trigonometric Functions. |