Section 5.6 Sinusoidal Graphs
This section contains analysis of sine and cosine as periodic functions.
Definition.
If
f
(
t
) =
A
sin
ωt
and
g
(
t
) =
A
cos
ωt
, then the number

A

is called the
amplitude
of
f
and
g
, and the number
T
=
2
π
ω
,
ω >
0, is called the
period
of
f
and
g
.
• 
A

Amplitude
of
A
sin
ωt
,
A
cos
ωt
•
T
=
2
π
ω
(
ω >
0)
Period
of
A
sin
ωt
,
A
cos
ωt
Note. The above formula for the period can be obtained as follows. The functions
f
(
t
) = sin
ωt
and
g
(
t
) = cos
ωt
complete one full cycle when
ωt
changes from
ωt
= 0 to
ωt
= 2
π
. This change in
ωt
occurs when
t
changes from
t
= 0 to
t
=
2
π
ω
.
Example.
Find the amplitude and period of
y
=

2 sin 3
t
.
Solution.
y
=

2 sin 3
t
is of form
y
=
A
sin
ωt
with
A
=

2 and
ω
= 3. Hence the amplitude is

A

= 2 and the period is
T
=
2
π
3
. The graph of this function is shown below.
–2
–1
0
1
2
y
–3
–2
–1
1
2
3
t
y
=

2 sin3
t
Example.
Find the amplitude and period of
y
= 3 cos(

2
t
).
Solution. Write
y
= 3 cos(

2
t
) in the form
y
= 3 cos(2
t
). (The last step is permissible because the
cosine is an even function.) The amplitude is

A

= 3 and the period
T
=
2
π
2
=
π
. The graph of this
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 Fall '11
 Kutter
 Algebra, Sin, Periodic function, ωt

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