Graphics Reference
In-Depth Information
.We
may now
define A
(
1)
and
A
as thecosinefunction, so that
cosine is a continuous and monotonic decreasing bijection with domain
[0,
] and range[
1, 1]. On thesamedomain wedefinethesine
function by
sin
x
(1
cos
x
).
Thefunction
A
is usually called arccos, and sometimes cos
Cosine and sine
51 Find thevalus of cos
x
and sin
x
when
x
0,
and
.
52 Usetheequation
A
(cos
x
)
x
to provethat cos
x
sin
x
for
0
x
. Usethedefinition of sineto provethat sin
x
cos
x
for
.
53 Sketch the graphs of cosine and sine on the domain [0,
]. For
x
0
x
x
).
Sketch the graphs of cosine and sine on the domain [0, 2
].
Verify that sin
x
cos
x
1 on [0, 2
], and that cos
x
sin
x
on (
,2
).
54 Provethat cos
sin
0, by applying theMean Value
Theorem to (cos
x
cos
)/(
x
), so that cos
x
sin
x
on
(0, 2
). Provelikewisethat sin
cos
1.
For any integer
k
, wenow
define
sineand cosinefor
2
k
x
2(
k
1)
by
cos
x
2
,
define
cos
x
cos(2
x
) and sin
x
sin(2
) and
sin
x
sin(
x
2
k
).
55 Provethat sin
x
cos
x
1 for all real
x
.
Provethat cos
x
sin
x
and sin
x
cos
x
, except possibly when
x
2
k
.
Usethemthod of qn 54 to provethat cos
0
sin 0
0 and
sin
0
cos 0
1, so that the formulae for the derived functions
hold for all values of
x
.
56 Define
f
(
x
)
sin(
a
x
) · cos
x
cos(
a
x
) · sin
x
. Provethat
f
cos(
x
2
k
0 for all
x
. Use the Mean Value Theorem to prove that
f
(
x
)
sin
a
. Deduce the formula for sin (
x
y
).
57 Usethemthod of qn 56 to provethat
cos (
x
y
)
cos
x
· cos
y
sin
x
· sin
y
.
58 Provethat cos 2
x
cos
x
sin
x
2 cos
x
1
1
2 sin
x
.
Provealso that sin 2
x
(
x
)
2 sin
x
· cos
x
.
