Graphing Secant and Cosecant
Finally, we will sketch the graphs of the secant and cosecant functions. These functions are easy to
sketch if we use their reciprocal functions, cosine and sine, as guides. (For this reason, you don’t need a
separate set of rules for transforming secant and cosecant.)
1
. The secant function has a vertical asymptote wherever the cosine function has
cos x
an x-intercept. At the point where cosine has its maximum value y  1 (or minimum value y  1 ),
First, y  sec x 
the values of the secant and cosine are the same. The graphs below show the secant function, first
graphed along with the cosine function, and then by itself.
4
4
3
3
2
2
1
1
3
2
2
-1
2
-2
-2
-3
-3
-4
-4
3
2
2
-1
Similarly, we can use the sine function as a guide when we graph y  csc x 
2
1
.Below, we show the
sin x
cosecant function, first graphed along with the sine function and then by itself.
4
4
3
3
2
2
1
1
-1
2
3
2
2
-1
-2
-2
-3
-3
-4
-4
2
3
2
2
1
y  2  csc(x) .
4
EXAMPLE 3: Sketch one period of the graph of
Solution: The graph will look exactly like the graph of the cosecant function, except that it will have
been shrunk by the value , the period is
, and the graph is shifted 2 units down. To sketch the
graph, we will first sketch the graph of y  2 
1
sin(x) , and use this graph as a guide. Reference
4
points on the graph are the points where the sine curve crosses the horizontal axis y = −2, where the
cosecant curve has vertical asymptotes, and the maximum and minimum points on the sine curve,
where the cosecant and sine curves touch. The graphs below show the sine curve by itself, the cosecant
and sine curves together, and finally the cosecant curve by itself.
EXAMPLE 4: Use your graphing calculator to sketch the graphs of y  2 
1
cos(2 x) and
2
1
y  2  sec( 2 x) in the same window.
2
Solution: After you’ve checked that your calculator is in radian mode, go to Y =, and, for y1 = , type
2 + 1/2*COS(2X), and, for y2 = , type 2 + 1/2*1  COS(2X). Go to ZOOM and then ZTRIG to get a good
view of both the cosine and corresponding secant functions. Do the asymptotes for the secant function
show up?