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Mathematics
Problem
The derivatives of
h
(
x
)
=
sin
x
h(x)=\sin x
h
(
x
)
=
sin
x
repeat every four orders. Identify the remainder
r
∈
{
0
,
1
,
2
,
3
}
r\in\{0,1,2,3\}
r
∈
{
0
,
1
,
2
,
3
}
such that
h
(
n
)
(
x
)
=
−
cos
x
h^{(n)}(x)=-\cos x
h
(
n
)
(
x
)
=
−
cos
x
when
n
≡
r
(
m
o
d
4
)
n\equiv r\pmod 4
n
≡
r
(
mod
4
)
, and briefly explain the pattern that justifies your answer.
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