4.6 Graphs of Other
Trigonometric FUNctions
How can I sketch the graphs of all of
the cool quadratic FUNctions?
Graph of the tangent FUNction
• The tangent FUNction is odd and periodic with
period π.
• As we saw in Section 2.6, FUNctions that are
fractions can have vertical asymptotes where
the denominator is zero and the numerator is
not.
sin x
tan
x

y  tan x
• Therefore, since
cos x , the graph of

will have vertical asymptotes at  2 n , where
2
n is an integer.
Let’s graph y = tan x.
• The tangent graph is so much easier to work
with then the sine graph or the cosine graph.
– We know the asymptotes.
– We know the x-intercepts.
y




x
















y = 2 tan (2x)
• Now, our period will be

b


2
• Additionally, the graph will get larger twice as
quickly.

• The asymptotes will be at  4
• The x-intercept will be (0,0)
y




x










x
y   tan  
2
• The period is 2π.
• The asymptotes are at ±π.
• The x-intercept is (0,0).
y




x
















Graph of a Cotangent FUNction
• Like the tangent FUNction, the cotangent
FUNction is
– odd.
– periodic.
– has a period of π.
• Unlike the tangent FUNction, the cotangent
FUNction has
– asymptotes at period πn.
y = cot x
• The asymptotes are at ±πn.


,
0
• There is an x-intercept at  2 
y




x
















y = -2 cot (2x)
• The period is

2
• There is an x-intercept at
• There is an asymptote at


 ,0 
 4 

2
y




x








Graphs of the Reciprocal FUNctions
• Just a reminder
– the sine and cosecant FUNctions are reciprocal
FUNctions
– the cosine and secant FUNctions are reciprocal
FUNctions
• So….
– where the sine FUNction is zero, the cosecant
FUNction has a vertical asymptote
– where the cosine FUNction is zero, the secant
FUNction has a vertical asymptote
• And…
– where the sine FUNction has a relative minimum,
the cosecant FUNction has a relative maximum
– where the sine FUNction has a relative maximum,
the cosecant FUNction has a relative minimum
– the same is true for the cosine and secant
FUNctions
• Let’s graph y = csc x
y




x
















Now, let’s graph y = sec x
y




x
















Now, you try your own….
• Just graph the FUNction as if it were a sine or
cosine FUNction, then make the changes we
have already made.
x
y  2 csc  
2
y  sec  x  

y




x
















y




x
















Damped Trigonometric Graphs (Just
for Fun!)
• Some FUNctions, when multiplied by a sine or
cosine FUNction, become damping factors.
• We use the properties of both FUNctions to
graph the new FUNction.
• For more fun on damping FUNctions, please
read p 339 in your textbook.
• For a nifty summary of the trigonometric
FUNctions, please check out page 340.
• As a matter of fact, I would make sure I
memorized all of the information on page 340.
Writing About Math
• Please turn to page 340 and complete the
Writing About Math – Combining
Trigonometric Functions.
• You may work with your group.
• This activity is due at the end of the class.