Graphics Reference
In-Depth Information
2
A
(0).
Definition A
(
1)
.
A
cosine.
Theorem
cos: [0,
qn 50
A
(
1)
1, 1]
qn 51 cos 0
1, cos
0, cos
1.
Definition
sin
x
]
[
].
cos
x
cos(2
x
)on(
,2
].
sin
x
sin(2
x
)on(
,2
].
cos
x
cos(
x
2
k
)on[2
k
,2(
k
1)
].
sin
x
sin(
x
2
k
)on[2
k
,2(
k
1)
].
Theorem
sin 0
(1
cos
x
) on [0,
0, sin
1, sin
0.
qns 52, 55,
cos
x
sin
x
, sin
x
cos
x
.
x
x
56, 57
sin
1.
sin(
x
y
)
sin
x
· cos
y
cos
x
· sin
y
.
cos(
x
y
)
cos
sin
x
· sin
y
.
Definition
tan
x
sin
x
/cos
x
provided
x
(
k
cos
x
· cos
y
)
.
T
heorem
tan
x
1
tan
x
.
qn 59
tan: (
,
)
R is a continuous bijection and
strictly increasing.
Historical note
with
n
, for
n
0, 1, 2, . . ., 100 and for a host of larger
n
, interpolating to build
up a table of log sines to seven significant figures. Such 'logarithms'
matching the terms of a geometric progression with those of an
arithmetic progression, satisfy log
a
In 1614, John Napier published tables matching 10
(1
10
)
log
b
log
c
log
d
ab
cd
,
and provide some of the advantages of modern logarithms for
computational purposes. Napier's achievement was the more
remarkable when one realises that neither the notion nor the notation
of exponents was developed until some 20 years later. Briggs had
discussions with Napier and then constructed tables (of common
logarithms) in which log 1 was 0 and log 10 was 1. These tables were
first published in 1617.
In 1647, work by Gregory of St Vincent on the hyperbola showed
that a geometric progression along one asymptote produced strips
parallel to the other asymptote whose areas were equal. In 1649,
Anthony de Sarasa claimed that this relationship was logarithmic.
In his
De Analysi
(written in 1669 and circulated among friends, but
not published until 1734) Newton had used term-by-term integration of
the series for 1/(1
x
) (which heobtained by long division) to find the
series for log(1
x
), and then developed a method of repeated
successive approximations to calculate the inverse function of a power
