Rewriting
B
cos(
ω
(
t

t
0
)) +
C
sin(
ω
(
t

t
0
))
as
A
cos(
ω
(
t

t
1
))
In several important applications, we will ﬁnd that we get solutions to dif
ferential equations in the form
B
cos(
ω
(
t

t
0
)) +
C
sin(
ω
(
t

t
0
))
where
B
,
C
and
T
0
are constants. In order to be able to clearly visualize and
easily understand the behavior of such functions, it is very useful to be able
to write them in the form
A
cos(
ω
(
t

t
1
))
In that form it is easy to see what the maximum and minimum values are,
and also for which values of
t
the maximum and the minimum are reached.
This rewriting is based on a trigonometric identity:
cos(
θ

φ
) = cos(
φ
) cos(
θ
) + sin(
φ
)sin(
θ
)
We can use that identity to try to match
Acos
(
θ

φ
) with
B
cos(
θ
)+
C
sin(
θ
).
We see that
Acos
(
θ

φ
) =
A
cos(
φ
)cos(
θ
) +
A
sin(
φ
)sin(
θ
)
,
so if we can solve
A
cos(
φ
) =
B
and
A
sin(
φ
) =
C
We will have
Acos
(
θ

φ
) =
B
cos(
θ
) +
C
sin(
θ
)
.
We can solve for
A
and
φ
by looking at a right triangle with base
B
, and
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 Spring '08
 INDIK
 Differential Equations, Trigonometry, Equations, Sin, Cos

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