4.4 Graphs of the Secant and Cosecant Functions Recall that sine and cosine have period 2 . Since secant and cosecant are the reciprocals of cosine and sine, respectively, they also have period 2 . We first sketch the graph of secant. Since it has period 2 , we need only sketch the graph on any interval of length 2 and then repeat the pattern to the left and to the right. Since sec
what can be said of the function values sec as cos gets close to zero, since division by zero is undefined. To see this we make a table of values and see what happens to sec as x approaches 1.570796 …. since cos
0. From the table of values we can see that sec
left of . Since sec
∞
the line ∞
through values of x to the is a vertical asymptote for the graph of sec . As a matter of fact, where ever the cosine function crosses the x‐axis there will be a vertical asymptote for the graph of the secant function. To graph the secant function we take reciprocals of the y‐coordinates of the points on the graph of the cosine function. 1. Graph sec . Solution: First graph cos . To graph the cosecant function we take reciprocals of the y‐coordinates of the points on the graph of the sine function and graph vertical asymptotes at the x‐intercepts of the sine function. 2. Graph csc 3. 3 sec
The period is as a guide. 8 . The length of each subinterval is 2 . Graph 3 cos
4. csc
Solution: Graph sin
subinterval is . One cycle occurs in the interval found by solving: 0
4
,
,
2 4
2
4
The endpoints of the subintervals are: , 2 . The length of each as a guide. The period is ,
9
4
4
5. 2 sec
Solution: Graph 2 cos
subinterval is . The interval in which one cycle occurs is found by solving the equations 4 . The length of each . The period is 1
2
1
2
3
1
2
1
2
1
2
0
3
2
3
2
3
3
2
2 3
5
3
10
3
The endpoints of the subintervals are: 2
4
,
,
,
3
3
3
,
7
3
6. csc 2
sin 2
Graph as a guide. The period is . The length of each subinterval is . The interval in which one cycle occurs is found by solving the equations: 2
0
2
2 2
2
2
2
The endpoints of the subintervals are: ,
2
4
,
4
4
,0
4
,
4
4