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1: 4.42 Solution of Triangles

… ►

4.42.5

c^{2}=a^{2}+b^{2}-2ab\cos C,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $a$ : length, $b$ : base, $c$ : length and $C$ : angle
A&S Ref:: 4.3.148
Permalink:: http://dlmf.nist.gov/4.42.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(ii), §4.42 and Ch.4

►

4.42.6

a=b\cos C+c\cos B

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $a$ : length, $b$ : base, $c$ : length, $B$ : angle and $C$ : angle
A&S Ref:: 4.3.148
Permalink:: http://dlmf.nist.gov/4.42.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(ii), §4.42 and Ch.4

… ►

4.42.8

\cos a=\cos b\cos c+\sin b\sin c\cos A,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $A$ : angle, $a$ : arc length, $b$ : arc length and $c$ : arc length
A&S Ref:: 4.3.149
Permalink:: http://dlmf.nist.gov/4.42.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.9

\frac{\sin A}{\sin a}=\frac{\sin B}{\sin b}=\frac{\sin C}{\sin c},

ⓘ

Symbols:: $\sin\NVar{z}$ : sine function, $A$ : angle, $B$ : angle, $C$ : angle, $a$ : arc length, $b$ : arc length and $c$ : arc length
A&S Ref:: 4.3.149
Permalink:: http://dlmf.nist.gov/4.42.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

►

4.42.10

\sin a\cos B=\cos b\sin c-\sin b\cos c\cos A,

ⓘ

Symbols:: $\cos\NVar{z}$ : cosine function, $\sin\NVar{z}$ : sine function, $A$ : angle, $B$ : angle, $a$ : arc length, $b$ : arc length and $c$ : arc length
Permalink:: http://dlmf.nist.gov/4.42.E10
Encodings:: pMML, png, TeX
See also:: Annotations for §4.42(iii), §4.42 and Ch.4

…

2: 22.18 Mathematical Applications

… ►

§22.18(i) Lengths and Parametrization of Plane Curves

►

Ellipse

… ►The arc length

l(u)

in the first quadrant, measured from

u=0

, is … ►

Lemniscate

… ►The arc length

l(r)

, measured from

\phi=0

, is …

3: 19.30 Lengths of Plane Curves

§19.30 Lengths of Plane Curves

►

§19.30(i) Ellipse

… ►The length of the ellipse is … ►

§19.30(ii) Hyperbola

… ►The perimeter length

P

of the lemniscate is given by …

4: 32.14 Combinatorics

… ►With

1\leq m_{1}<\cdots<m_{n}\leq N

\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})

is said to be an increasing subsequence of

\boldsymbol{\pi}

of length

n

when

\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})

. Let

\ell_{N}(\boldsymbol{\pi})

be the length of the longest increasing subsequence of

\boldsymbol{\pi}

. … ►

32.14.1

\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),

ⓘ

Symbols:: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function
Permalink:: http://dlmf.nist.gov/32.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

…

5: 23.23 Tables

… ►The values are tabulated on the real and imaginary

z

-axes, mostly ranging from 0 to 1 or

i

in steps of length 0. …

6: 26.13 Permutations: Cycle Notation

… ►Cycles of length one are fixed points. … ►An element of

\mathfrak{S}_{n}

with

a_{1}

fixed points,

a_{2}

cycles of length

2,\ldots,a_{n}

cycles of length

n

, where

n=a_{1}+2a_{2}+\cdots+na_{n}

, is said to have cycle type

{\left(a_{1},a_{2},\ldots,a_{n}\right)}

. … ►A transposition is a permutation that consists of a single cycle of length two. …A permutation that consists of a single cycle of length

k

can be written as the composition of

k-1

two-cycles (read from right to left): …

7: 27.16 Cryptography

… ►To code a message by this method, we replace each letter by two digits, say

A=01

B=02

\dots

Z=26

, and divide the message into pieces of convenient length smaller than the public value

n=pq

. …

8: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ►

M_{2}

is the number of permutations of

\{1,2,\ldots,n\}

with

a_{1}

cycles of length 1,

a_{2}

cycles of length 2,

\dotsc

, and

a_{n}

cycles of length

n

: …

9: 3.1 Arithmetics and Error Measures

… ►Computer algebra systems use exact rational arithmetic with rational numbers

p/q

, where

p

and

q

are multi-length integers. During the calculations common divisors are removed from the rational numbers, and the final results can be converted to decimal representations of arbitrary length. …

10: 7.20 Mathematical Applications

… ►Then the arc length between the origin and

P(t)

equals

t

, and is directly proportional to the curvature at

P(t)

, which equals

\pi t

. …