4.6 Graphing y = sec(x), y = csc(x), y = tan(x) and y = cot(x)
Graphing Secant and Cosecant
Remember: y = sec(x) =
1
cos(x)
and
y = csc(x) =
1
sin(x)
This means the y‐values on the graphs of secant are the reciprocals of the y‐
values on the graph of cosine (same for cosecant and sine).
Using a dotted graph of cosine, we can get a graph of secant.
When taking reciprocals:
The reciprocal of 1 is 1 and the reciprocal of –1 is –1. This means the graph
of secant will ‘touch’ cosine’s graph at the amplitude max and min points.
The reciprocal of 0 is undefined. This means the graph of secant will have
VERTICAL ASYMPTOTES at the x‐intercepts of cosine’s graph.
The graph of y = sec(x) (made by using a dotted cosine graph for reference)
The graph of y = csc(x) (made by using a dotted sine graph for reference)
90% of the work in making a secant or cosecant graph is done by making the
dotted reference graph of its reciprocal: cosine as a reference for secant and
sine as a reference for cosecant.
ex) Graph two full periods of y = 3csc( π3 x) .
Dotted reference graph will be y = 3sin( π3 x)
ex) Graph two full periods of y = sec ( 4 x − π )
Dotted reference graph will be y = cos ( 4 x − π ) .
Graphing Tangent and Cotangent
The graphs of Tangent and Cotangent are unlike the other 4 functions.
Period = π /B (instead of 2π /B like the other 4)
The range of Tangent and Cotangent is all real numbers.
y = tan(x)
y = cot(x)
One full period − π2 < x < π2
Graph has x‐int. at x = 0
One full period 0 < x < π
Graph has a V.A. at x = 0
Since these functions have infinite range, the only real effect that a coefficient in
front would have on the graph is its steepness. The graphs usually cross the x‐axis
at a slope of +1 for tangent and –1 for cotangent. The coefficients in front would
stretch or shrink these slopes.
ex) Graph three full periods of y = 4 tan( 3x )
There are only 3 critical points needed for a period of tangent and cotangent:
1st: V.A. 2nd: x‐intercept 3rd: V.A.
The distance between the vertical asymptotes is the period distance π /B
Period distance =
Increment = ½ period =
REMEMBER Tangent’s graph has an x‐intercept at x = 0.
ex) Graph three full periods of y = cot(π x)
Period distance =
Increment = ½ period =
REMEMBER Cotangent’s graph has a vertical asymptote at x = 0.
When shifting their graphs you’ll need to remember what role x = 0 plays for each
function.
ex) Graph three full periods of y = − tan( 2 x + π )
Period distance =
Set the argument = 0 to determine the value which will play the role of x = 0.
Shifted “0” point:
You’ll need to position the y‐axis after locating the critical points.