Graphing the Trigonometric Functions
•
Section 5.1
105
x
y
0
−
8
−
6
−
4
−
2
2
4
6
8
π
4
π
2
3
π
4
π
5
π
4
3
π
2
7
π
4
2
π
−
π
4
−
π
2
−
3
π
4
−
π
−
5
π
4
−
3
π
2
−
7
π
4
−
2
π
y
=
tan
x
Figure 5.1.6
Graph of
y
=
tan
x
Recall that the tangent is positive for angles in QI and QIII, and is negative in QII and
QIV, and that is indeed what the graph in Figure 5.1.6 shows. We know that tan
x
is not
defined when cos
x
=
0, i.e. at odd multiples of
π
2
:
x
=±
π
2
,
±
3
π
2
,
±
5
π
2
, etc. We can figure out
what happens
near
those angles by looking at the sine and cosine functions. For example,
for
x
in QI near
π
2
, sin
x
and cos
x
are both positive, with sin
x
very close to 1 and cos
x
very
close to 0, so the quotient tan
x
=
sin
x
cos
x
is a positive number that is very large. And the closer
x
gets to
π
2
, the larger tan
x
gets. Thus,
x
=
π
2
is a
vertical asymptote
of the graph of
y
=
tan
x
.
Likewise, for
x
in QII very close to
π
2
, sin
x
is very close to 1 and cos
x
is negative and very
close to 0, so the quotient tan
x
=
sin
x
cos
x
is a negative number that is very large, and it gets
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 Fall '11
 Dr.Cheun
 Calculus, Angles, Sin, Tan, QII

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