Graphics Reference
In-Depth Information
Tangent
59 Define tan
x
sin
x
/cos
x
, except where
x
(
k
, and prove
that tan
x
1/cos
x
tan
x
1. Show that thetangent function
is a monotonic increasing bijection (
)
,
)
R
.
60 Define arctan:
) as thefunction invrseto tan, so that
arctan (tan
x
)
x
. Explain why arctan is differentiable and why
arctan
R
(
,
x
1/(1
x
).
61 Usetheequation
1
(
t
)
1
1
(
t
)
(
t
)
...
(
t
)
)
,
(
t
which is valid for all
t
, and the Fundamental Theorem of Calculus,
to show that
x
3
x
5
...
(
1)
x
2
n
t
arctan
x
x
1
(
1)
dt
.
1
t
Check the terms of this Maclaurin expansion with 1.8(ii).
Provethat
t
1
t
t
dt
x
(
1)
dt
2
n
1
.
Check that this last expression gives a null sequence when
x
1
and determine for what values of
x
the power series developed here
is a valid expansion of arctan
x
.
62 Provethat
1
3
1
5
1
7
1
1
...
(
1)
1
...
2
n
Summary
-
Circular or trigonometric functions
Theorem
Thearc length on theunit circlefrom (1, 0) to
qn 48
(
x
,
(1
x
)),
1
x
1,
dy
A
(
x
)
)
.
(1
y
Theorem
Thefunction
A
:[
1, 1]
[0,
A
(
1)] is
continuous, decreasing and
qn 49
1
A
(
x
)
)
on (
1, 1).
x
(1
