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.
The cosecant function \( csc (\theta) \) is defined as
\( \csc(\theta) = \dfrac{r}{y} \) , where \( r \) is the distance from O to M and is given by \( r = \sqrt{x^2+y^2} \).
An examination of the definiton of the cosecant gives a relationship between \( csc (\theta) \) and \( sin (\theta) \) as follows
\( \csc(\theta) = \dfrac{r}{y} = \dfrac{1}{\sin(\theta)}\)
Note that
1) \( \csc(\theta+2\pi) = \dfrac{1}{\sin(\theta+2\pi)} = \dfrac{1}{\sin(\theta)}= \csc(\theta)\)
and therefore \( \csc(\theta) \) is a periodic function whose period is equal to \( 2\pi \).
2) \( \csc(-\theta) = \dfrac{1}{\sin(-\theta)} = \dfrac{1}{-\sin(\theta)} = - \dfrac{1}{\sin(\theta)} = - \csc(\theta)\)
and therefore \( \csc(\theta) \) is an odd function and its graph is symmetric with respect to the origin of a rectangular system of coordinates.
We now use a unit circle to find \( \sin(\theta)\) and hence \( \csc(\theta)\) over one period extending from \( \theta = 0 \) to \( \theta = 2\pi \).
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 quadrantal angles angles \( 0, \dfrac{\pi}{2} , \pi , \dfrac{3\pi}{2} , 2\pi \) and the corresponding values of \( \sin(\theta)\) and \( \csc (\theta) = \dfrac{1}{\sin (\theta)} \) on a table as shown below.
| \( \theta \) | \( \sin(\theta) \) | \( \csc (\theta) = \dfrac{1}{\sin (\theta)} \) |
| \( 0 \) | \( 0 \) | undefined |
| \( \dfrac{\pi}{2} \) | \( 1 \) | 1 |
| \( \pi \) | \( 0 \) | undefined |
| \( \dfrac{3\pi}{2} \) | \( -1 \) | -1 |
| \( 2\pi \) | \( 0 \) | undefined |
\( \csc(\theta)\) 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 \( \theta = -0.1 \)
| \( \theta \) | \( \csc(\theta) \) |
| \( -0.1 \) | \( -10.01668613 \) |
| \( -0.01 \) | \( -100.0016667 \) |
| \( -0.001 \) | \( -1000.000167 \) |
| \( -0.000001 \) | \( -1000000 \) |
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 \( \theta = 0.1 \)
| \( \theta \) | \( \csc(\theta) \) |
| \( 0.1 \) | \( 10.01668613 \) |
| \( 0.01 \) | \( 100.0016667 \) |
| \( 0.001 \) | \( 1000.000167 \) |
| \( 0.000001 \) | \( 1000000 \) |
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 \( \csc x \).
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Properties of csc x
1) csc x has a period equal to \( 2\pi \).
2) \( \csc(x) \) has vertical asymptotes at all values of \( x = n\pi \) , \( n \) being any integer.
3) The domain of \( \csc(x) \) is the set of all real numbers except \( x = n\pi \) , \( n \) being any integer.
4) The range of \( \csc(x) \) is given by: \( (-\infty , -1] \cup [1, +\infty) \)
5) \( \csc(x) \) is an odd function and its graph is symmetric with respect to the origin of the system of axes.
Interactive Tutorial on the Cosecant Function csc x of the General Form
A 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 range? If yes how?
2 - Set a = 1, c = 0, d = 0 and change b. For each value of b find the period from the graph and compare it to \( \dfrac{2\pi}{|b|} \), formula of the period of \( f(x) = a \csc( b x + c ) + d \). How does b affect the graph of f(x)? How does it affect the vertical asymptotes?
3 - Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to positive large values. Take note of the shift, is it left or right, and compare it to \( - c / b\), formula of the phase shift of \( f(x) = a \csc( b x + c ) + d \).
4 - Set a = 1, b = 1, d = 0 and change c starting from zero going slowly to negative smaller values. Take note of the shift, is it left or right, and compare it to \( - c / b\).
5 - Repeat steps 4 and 5 above for b = 2, 3 and 4.
6 - Set a, b and c to non zero values and change d. What is the direction of the shift of the graph? How is the range of the function affected?
7 - Which of the parameters affect the positions of the vertical asymptotes? Explain analytically.
8 - Which of the parameters affect the domain of the function? Explain analytically.
9 - Which of the parameters affect the range of the cosecant function? Explain analytically.
More references on the trigonometric functions
Properties of Trigonometric Functions.Trigonometric Functions.
