aarjaneiro/planetmath huggingface dataset:
definitions in trigonometry

definitions in trigonometry

Informal definitions

Given a

triangle

A

⁢

B

⁢

C

with a signed angle

x

at

A

and a

right angle

at

B

, the ratios

B

⁢

C

A

⁢

C

A

⁢

B

A

⁢

C

B

⁢

C

A

⁢

B

are dependent only on the angle

x

, and therefore define

functions

,
denoted by

sin

⁡

x

cos

⁡

x

tan

⁡

x

respectively, where the names are short for

sine, cosine

and

tangent

. Their

inverses

are rather less important,
but also have names:

cot

⁡

x

=

A

⁢

B

B

⁢

C

=

1

tan

⁡

x

⁢

(cotangent)

csc

⁡

x

=

A

⁢

C

B

⁢

C

=

1

sin

⁡

x

⁢

(cosecant)

sec

⁡

x

=

A

⁢

C

A

⁢

B

=

1

cos

⁡

x

⁢

(secant)

From Pythagoras’s theorem we have

cos

2

⁡

x

+

sin

2

⁡

x

=

1

for all (real)

x

.
Also it is “clear” from the diagram at left that functions

cos

and

sin

are

periodic

with period

2

⁢

π

. However:

Formal definitions

The above definitions are not fully rigorous, because we have not defined
the word

angle

. We will sketch a more rigorous approach.

The

power series

∑

n

=

0

∞

x

n

n

!

converges uniformly on

compact subsets

of

ℂ

and its sum,
denoted by

exp

⁡

(

x

)

or by

e

x

, is therefore an

entire function

of

x

,
called the

exponential function

.

f

⁢

(

x

)

=

exp

⁡

(

x

)

is the unique solution of the boundary value problem

f

⁢

(

0

)

=

1

f

′

⁢

(

x

)

=

f

⁢

(

x

)

on

ℝ

.
The sine and

cosine functions

, for real

arguments

, are defined in terms
of

exp

, simply by

exp

⁡

(

i

⁢

x

)

=

cos

⁡

x

+

i

⁢

(

sin

⁡

x

)

.

Thus

cos

⁡

x

=

1

-

x

2

2

!

+

x

4

4

!

-

x

6

6

!

+

⋯

sin

⁡

x

=

x

1

!

-

x

3

3

!

+

x

5

5

!

-

⋯

Although it is not self-evident,

cos

and

sin

are

periodic functions

on
the

real line

, and have the same period. That period is denoted by

2

⁢

π

.

Title

definitions in trigonometry

Canonical name

DefinitionsInTrigonometry

Date of creation

2013-03-22 13:55:08

Last modified on

2013-03-22 13:55:08

Owner

Daume (40)

Last modified by

Daume (40)

Numerical id

10

Author

Daume (40)

Entry type

Definition

Classification

msc 26A09

Related topic

Trigonometry

Related topic

Sinusoid

Related topic

ComplexSineAndCosine

Related topic

ExampleOnSolvingAFunctionalEquation

Related topic

DerivativesOfSineAndCosine

Related topic

AdditionFormulasForSineAndCosine

Related topic

AdditionFormulaForTangent

Related topic

GoniometricFormulae

Related topic

OsculatingCurve

Defines

sine

Defines

cosine

Defines

exponential

Defines

tangent

Defines

cotangent

Defines

secant

Defines

cosecant

Defines

trigonometric function

Generated on Fri Feb 9 19:46:53 2018 by

LaTeXML
derivation of half-angle formulae for tangent

derivation of half-angle formulae for tangent

Start with the angle duplication formula

tan

⁡

(

x

)

=

2

⁢

tan

⁡

(

x

/

2

)

1

-

tan

2

⁡

(

x

/

2

)

.

Cross-multiply and move terms around:

tan

⁡

(

x

)

⁢

tan

2

⁡

(

x

/

2

)

+

2

⁢

tan

⁡

(

x

/

2

)

=

tan

⁡

(

x

)

Divide by

tan

⁡

(

x

)

:

tan

2

⁡

(

x

/

2

)

+

2

⁢

tan

⁡

(

x

/

2

)

tan

⁡

x

=

1

Add

1

/

tan

2

⁡

(

x

)

to both sides:

tan

2

⁡

(

x

/

2

)

+

2

⁢

tan

⁡

(

x

/

2

)

tan

⁡

x

+

1

tan

2

⁡

(

x

)

=

1

+

1

tan

2

⁡

(

x

)

Complete

the square (

http://planetmath.org/CompletingTheSquare

):

(

tan

⁡

(

x

/

2

)

+

1

tan

⁡

(

x

)

)

2

=

1

+

1

tan

2

⁡

(

x

)

Take a

square root

and move a term to obtain the

half-angle formula

:

tan

⁡

(

x

/

2

)

=

1

+

1

tan

2

⁡

(

x

)

-

1

tan

⁡

(

x

)

To derive the other forms of the formula, we start by substituting

sin

⁡

(

x

)

/

cos

⁡

(

x

)

for

tan

⁡

(

x

)

:

tan

⁡

(

x

/

2

)

=

1

+

cos

2

⁡

(

x

)

sin

2

⁡

(

x

)

-

cos

⁡

(

x

)

sin

⁡

(

x

)

Put the stuff inside the square root over a common

denominator

:

sin

2

⁡

(

x

)

+

cos

2

⁡

(

x

)

sin

2

⁡

(

x

)

-

cos

⁡

(

x

)

sin

⁡

(

x

)

Recall that

sin

2

⁡

(

x

)

+

cos

2

⁡

(

x

)

=

1

. Hence, we may get rid
of the square root:

1

sin

⁡

x

-

cos

⁡

(

x

)

sin

⁡

(

x

)

Putting the terms over a common denominator, we obtain our formula:

tan

⁡

(

x

/

2

)

=

1

-

cos

⁡

(

x

)

sin

⁡

(

x

)

To obtain the next formula, multiply both numerator and denominator
by

1

+

cos

⁡

(

x

)

:

(

1

-

cos

⁡

(

x

)

)

⁢

(

1

+

cos

⁡

(

x

)

)

sin

⁡

(

x

)

⁢

(

1

+

cos

⁡

(

x

)

)

Multiply out the numerator and simplify:

1

-

cos

2

⁡

(

x

)

sin

⁡

(

x

)

⁢

(

1

+

cos

⁡

(

x

)

)

Note that the numerator equals

sin

2

⁡

(

x

)

:

sin

2

⁡

(

x

)

sin

⁡

(

x

)

⁢

(

1

+

cos

⁡

(

x

)

)

Cancel a common factor of

sin

⁡

(

x

)

to obtain the formula

tan

⁡

(

x

/

2

)

=

sin

⁡

(

x

)

1

+

cos

⁡

(

x

)

.

To obtain the last formula, multiply the previous two formulae:

tan

2

⁡

(

x

/

2

)

=

1

-

cos

⁡

(

x

)

sin

⁡

(

x

)

⋅

sin

⁡

(

x

)

1

+

cos

⁡

(

x

)

Cancel the common factor of

sin

⁡

(

x

)

:

tan

2

⁡

(

x

/

2

)

=

1

-

cos

⁡

(

x

)

1

+

cos

⁡

(

x

)

Take the square root of both sides to obtain the formula

tan

⁡

x

2

=

±

1

-

cos

⁡

x

1

+

cos

⁡

x

;

here the sign (

±

) has to be chosen according to the

quadrant

where the angle

x

2

is.

Title

derivation of half-angle formulae for tangent

Canonical name

DerivationOfHalfangleFormulaeForTangent

Date of creation

2013-03-22 17:00:19

Last modified on

2013-03-22 17:00:19

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

9

Author

rspuzio (6075)

Entry type

Derivation

Classification

msc 26A09

Related topic

TangentOfHalvedAngle

Generated on Fri Feb 9 19:46:56 2018 by

LaTeXML
derivatives of hyperbolic functions

derivatives of hyperbolic functions

In this entry we compute the

derivative

of the

hyperbolic functions

sinh

⁡

(

x

)

and

cosh

⁡

(

x

)

.

Recall that by definition:

sinh

⁡

(

x

)

:=

e

x

-

e

-

x

2

cosh

⁡

(

x

)

:=

e

x

+

e

-

x

2

.

Therefore:

d

d

⁢

x

⁢

sinh

⁡

(

x

)

=

d

d

⁢

x

⁢

(

e

x

-

e

-

x

2

)

=

1

2

⋅

d

d

⁢

x

⁢

(

e

x

-

e

-

x

)

=

1

2

⋅

(

e

x

-

(

-

e

-

x

)

)

=

e

x

+

e

-

x

2

=

cosh

⁡

(

x

)

.

Similarly

d

d

⁢

x

⁢

cosh

⁡

(

x

)

=

sinh

⁡

(

x

)

. Using the

quotient rule

, we compute the derivative of

tanh

⁡

(

x

)

=

sinh

⁡

(

x

)

cosh

⁡

(

x

)

:

d

d

⁢

x

⁢

tanh

⁡

(

x

)

=

cosh

2

⁡

(

x

)

-

sinh

2

⁡

(

x

)

cosh

2

⁡

(

x

)

=

1

cosh

2

⁡

(

x

)

where we have used the equality

cosh

2

⁡

(

x

)

-

sinh

2

⁡

(

x

)

=

1

.

Title

derivatives of hyperbolic functions

Canonical name

DerivativesOfHyperbolicFunctions

Date of creation

2013-03-22 14:32:18

Last modified on

2013-03-22 14:32:18

Owner

alozano (2414)

Last modified by

alozano (2414)

Numerical id

5

Author

alozano (2414)

Entry type

Derivation

Classification

msc 26A09

Generated on Fri Feb 9 19:46:58 2018 by

LaTeXML
derivatives of sine and cosine

derivatives of sine and cosine

The of the

derivatives of sine and cosine

is a bit simpler by using the

prosthaphaeresis formulas

sin

⁡

α

-

sin

⁡

β

=

2

⁢

sin

⁡

(

α

-

β

2

)

⁢

cos

⁡

(

α

+

β

2

)

,

(1)

cos

⁡

α

-

cos

⁡

β

=

-

2

⁢

sin

⁡

(

α

+

β

2

)

⁢

sin

⁡

(

α

-

β

2

)

.

(2)

Let

x

,

t

be any

real numbers

such that

t

≠

x

.  Then we obtain

sin

⁡

x

-

sin

⁡

t

x

-

t

=

2

⁢

sin

⁡

(

x

-

t

2

)

⁢

cos

⁡

(

x

+

t

2

)

x

-

t

=

sin

⁡

(

x

-

t

2

)

(

x

-

t

2

)

⋅

cos

⁡

(

x

+

t

2

)

⟶

1

⋅

cos

⁡

(

x

+

x

2

)

=

cos

⁡

x

,

as

t

→

x

.  Here we used the known limit

lim

u

→

0

⁡

sin

⁡

u

u

=

1

(see this entry (

http://planetmath.org/LimitOfDisplaystyleFracsinXxAsXApproaches0

)).

The

derivative

of cosine is calculated similarly:

cos

⁡

x

-

cos

⁡

t

x

-

t

=

-

2

⁢

sin

⁡

(

x

+

t

2

)

⁢

sin

⁡

(

x

-

t

2

)

x

-

t

=

-

1

⋅

sin

⁡

(

x

-

t

2

)

(

x

-

t

2

)

⋅

sin

⁡

(

x

+

t

2

)

⟶

-

1

⋅

1

⋅

sin

⁡

(

x

+

x

2

)

=

-

sin

⁡

x

,

as

t

→

x

.

Title

derivatives of sine and cosine

Canonical name

DerivativesOfSineAndCosine

Date of creation

2013-03-22 16:58:58

Last modified on

2013-03-22 16:58:58

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

10

Author

Wkbj79 (1863)

Entry type

Derivation

Classification

msc 26A09

Related topic

DerivativesOfSinXAndCosX

Related topic

LimitOfDisplaystyleFracsinXxAsXApproaches0

Related topic

DefinitionsInTrigonometry

Related topic

LimitRulesOfFunctions

Generated on Fri Feb 9 19:47:01 2018 by

LaTeXML
double angle identity

double angle identity

The

double angle identities

are

sin

⁡

(

2

⁢

x

)

=

2

⁢

sin

⁡

x

⁢

cos

⁡

x

(1)

cos

⁡

(

2

⁢

x

)

=

cos

2

⁡

x

-

sin

2

⁡

x

=

2

⁢

cos

2

⁡

x

-

1

=

1

-

2

⁢

sin

2

⁡

x

(2)

tan

⁡

(

2

⁢

x

)

=

2

⁢

tan

⁡

x

1

-

tan

2

⁡

x

(3)

These are all derived from their respective

trigonometric addition formulas

. For example,

sin

⁡

(

2

⁢

x

)

=

sin

⁡

(

x

+

x

)

=

sin

⁡

x

⁢

cos

⁡

x

+

cos

⁡

x

⁢

sin

⁡

x

=

2

⁢

sin

⁡

x

⁢

cos

⁡

x

The formula for cosine follows similarly, and the formula

tangent

is derived by taking the ratio of sine to cosine, as always.

The double angle identities can also be derived from the

de Moivre identity

.

Title

double angle identity

Canonical name

DoubleAngleIdentity

Date of creation

2013-03-22 12:14:31

Last modified on

2013-03-22 12:14:31

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

16

Author

Wkbj79 (1863)

Entry type

Theorem

Classification

msc 26A09

Classification

msc 33B10

Synonym

double-angle identity

Synonym

double angle formula

Synonym

double-angle formula

Synonym

double angle formulae

Synonym

double-angle formulae

Related topic

DeMoivreIdentity

Related topic

AngleSumIdentity

Related topic

AdditionFormulasForSineAndCosine

Generated on Fri Feb 9 19:47:02 2018 by

LaTeXML
fourth power

fourth power

The

fourth power

of a number

x

is the number obtained multiplying

x

by itself three times thus:

x

×

x

×

x

×

x

. It’s more commonly denoted as

x

4

. For example,

2

4

=

2

×

2

×

2

×

2

=

16

. Since the square of a

square number

is a fourth power,

x

2

⁢

x

2

=

x

2

+

2

=

x

4

, fourth powers are sometimes called

biquadratic numbers

. For example,

2

4

=

2

2

⁢

2

2

=

4

2

=

16

. The first few

integer

fourth powers are 1, 16, 81, 256, 625, 1296, 2401, etc., listed in A000290 of Sloane’s OEIS.

Any integer can be represented by the sum of at most 19 integer fourth powers (see Waring’s problem).

Euler’s

conjecture

was first disproven with fifth powers, but there are also

counterexamples

using fourth powers. Sloane’s A003828 lists the known integers

n

having solutions to

n

4

=

a

4

+

b

4

+

c

4

.

The fourth power of a

negative number

is always a positive number; the fourth root of a negative real number is a

complex number

a

+

b

⁢

i

with

|

a

|

=

|

b

|

and

a

≠

0

.

Title

fourth power

Canonical

name

FourthPower

Date of creation

2013-03-22 18:25:16

Last modified on

2013-03-22 18:25:16

Owner

CompositeFan (12809)

Last modified by

CompositeFan (12809)

Numerical id

5

Author

CompositeFan (12809)

Entry type

Definition

Classification

msc 26A09

Classification

msc 11A05

Synonym

biquadratic number

Generated on Fri Feb 9 19:47:12 2018 by

LaTeXML
hyperbolic functions

hyperbolic functions

The

hyperbolic functions

sinh

(

sinus hyperbolicus

) and

cosh

(

cosinus hyperbolicus

) with arbitrary complex argument

x

are defined as follows:

sinh

⁡

x

:=

e

x

-

e

-

x

2

,

cosh

⁡

x

:=

e

x

+

e

-

x

2

.

One can then also also define the

functions

tanh

(

tangens hyperbolica

) and

coth

(

cotangens hyperbolica

) in analogy to the definitions of

tan

and

cot

:

tanh

⁡

x

:=

sinh

⁡

x

cosh

⁡

x

=

e

x

-

e

-

x

e

x

+

e

-

x

,

coth

⁡

x

:=

cosh

⁡

x

sinh

⁡

x

=

e

x

+

e

-

x

e

x

-

e

-

x

.

We further define the

sech

and

csch

:

sech

⁡

x

:=

1

cosh

⁡

x

=

2

e

x

+

e

-

x

,

csch

⁡

x

:=

1

sinh

⁡

x

=

2

e

x

-

e

-

x

,

where

cosh

⁡

x

resp.

sinh

⁡

x

is not

0

.

Figure 1:

Graphs of the hyperbolic functions.

The hyperbolic functions are named in that way because the hyperbola

x

2

a

2

-

y

2

b

2

=

1

can be written in parametrical form with the equations:

x

=

a

⁢

cosh

⁡

t

,

y

=

b

⁢

sinh

⁡

t

.

This is because of the equation

cosh

2

⁡

x

-

sinh

2

⁡

x

=

1

.

There are also

addition formulas

which are like the ones for

trigonometric functions

:

sinh

⁡

(

x

±

y

)

=

sinh

⁡

x

⁢

cosh

⁡

y

±

cosh

⁡

x

⁢

sinh

⁡

y

cosh

⁡

(

x

±

y

)

=

cosh

⁡

x

⁢

cosh

⁡

y

±

sinh

⁡

x

⁢

sinh

⁡

y

.

The

Taylor series

for the hyperbolic functions are:

sinh

⁡

x

=

∑

n

=

0

∞

x

2

⁢

n

+

1

(

2

⁢

n

+

1

)

!

cosh

⁡

x

=

∑

n

=

0

∞

x

2

⁢

n

(

2

⁢

n

)

!

.

There are the following between the hyperbolic and the trigonometric functions:

sin

⁡

x

=

sinh

⁡

(

i

⁢

x

)

i

cos

⁡

x

=

cosh

⁡

(

i

⁢

x

)

.

Title

hyperbolic functions

Canonical name

HyperbolicFunctions

Date of creation

2013-03-22 12:38:27

Last modified on

2013-03-22 12:38:27

Owner

mathwizard (128)

Last modified by

mathwizard (128)

Numerical id

13

Author

mathwizard (128)

Entry type

Definition

Classification

msc 26A09

Related topic

UnitHyperbola

Related topic

ComplexTangentAndCotangent

Related topic

ParallelCurve

Related topic

HyperbolicAngle

Related topic

ExampleOfCauchyMultiplicationRule

Related topic

DerivationOfFormulasForHyperbolicFunctionsFromDefinitionOfHyperbolicAngle

Related topic

HeavisideFormula

Related topic

Catenary

Related topic

HyperbolicSineIntegral

Related topic

InverseGudermannia

Defines

sinh

Defines

cosh

Defines

tanh

Defines

coth

Defines

sech

Defines

csch

Defines

hyperbolic sine

Defines

hyperbolic cosine

Defines

hyperbolic tangent

Defines

hyperbolic cotangent

Defines

hyperbolic secant

Defines

hyperbolic cosecant

Generated on Fri Feb 9 19:47:24 2018 by

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integral related to arc sine

integral related to arc sine

We want to evaluate the

integral

∫

1

∞

(

arcsin

⁡

1

x

-

1

x

)

⁢

𝑑

x

.

(1)

Therefore we put an extra

variable

t

to the

integrand

and thus get the function

I

⁢

(

t

)

:=

∫

1

∞

(

arcsin

⁡

t

x

-

t

x

)

⁢

𝑑

x

,

and in to obtain a simpler integral, we

differentiate

it under the integral sign (

http://

planetmath

.org/DifferentiationUnderIntegralSign

), then integrate:

I

′

⁢

(

t

)

=

∫

1

∞

(

1

1

-

t

2

x

2

⋅

1

x

-

1

x

)

⁢

𝑑

x

=

∫

1

∞

(

1

x

2

t

2

-

1

⋅

1

t

-

1

x

)

⁢

𝑑

x

=

/

x

=

1

∞

⁡

[

ln

⁡

(

x

t

+

x

2

t

2

-

1

)

-

ln

⁡

x

]

=

/

x

=

1

∞

⁡

ln

⁡

1

+

1

-

t

2

x

2

t

=

ln

⁡

2

t

-

ln

⁡

1

+

1

-

t

2

t

=

ln

⁡

2

-

ln

⁡

(

1

+

1

-

t

2

)

The gotten

expression

implies, since

I

⁢

(

0

)

=

∫

1

∞

(

arcsin

⁡

0

-

0

)

⁢

𝑑

x

=

0

,  that

I

⁢

(

t

)

=

∫

0

t

[

ln

⁡

2

-

ln

⁡

(

1

+

1

-

t

2

)

]

⁢

𝑑

t

=

t

⁢

ln

⁡

2

-

∫

0

t

ln

⁡

(

1

+

1

-

t

2

)

⁢

𝑑

t

,

and consequently

I

⁢

(

1

)

=

ln

⁡

2

-

∫

0

1

ln

⁡

(

1

+

1

-

t

2

)

⁢

𝑑

t

=

ln

⁡

2

-

/

0

1

⁡

t

⁢

ln

⁡

(

1

+

1

-

t

2

)

-

∫

0

1

t

2

⁢

d

⁢

t

(

1

+

1

-

t

2

)

⁢

1

-

t

2

=

ln

⁡

2

-

∫

0

1

t

2

⁢

d

⁢

t

1

-

t

2

+

1

-

t

2

.

Here, the substitution (

http://planetmath.org/ChangeOfVariableInDefiniteIntegral

)

t

=

sin

⁡

u

helps, yielding

I

⁢

(

1

)

=

ln

⁡

2

-

∫

0

π

2

(

1

-

cos

⁡

u

)

⁢

𝑑

u

=

ln

⁡

2

-

π

2

+

1

.

Accordingly, we have the result

∫

1

∞

(

arcsin

⁡

1

x

-

1

x

)

⁢

𝑑

x

=

1

+

ln

⁡

2

-

π

2

.

For the convergence, see the French version of

http://en.

wikipedia

.org/wiki/Improper_integral

this article.

Title

integral related to arc sine

Canonical

name

IntegralRelatedToArcSine

Date of creation

2013-03-22 18:44:58

Last modified on

2013-03-22 18:44:58

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Example

Classification

msc 26A09

Related topic

SubstitutionNotation

Related topic

ArcSine

Related topic

Arcosh

Related topic

MethodsOfEvaluatingImproperIntegrals

Related topic

CyclometricFunctions

Generated on Fri Feb 9 19:47:27 2018 by

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logarithm

logarithm

Definition.

Three

real numbers

x

,

y

,

p

, with

x

,

y

>

0

and

x

≠

1

, are said to obey the
logarithmic relation

log

x

⁡

(

y

)

=

p

if they obey the corresponding

exponential

relation:

x

p

=

y

.

Note that by the monotonicity and

continuity

property

of the
exponential operation, for given

x

and

y

there exists a unique

p

satisfying the above relation. We are therefore able to says that

p

is the

logarithm

of

y

relative to the base

x

.

Properties.

There are a number of basic algebraic identities involving logarithms.

log

x

⁡

(

y

⁢

z

)

=

log

x

⁡

(

y

)

+

log

x

⁡

(

z

)

log

x

⁡

(

y

z

)

=

log

x

⁡

(

y

)

-

log

x

⁡

(

z

)

log

x

⁡

(

y

z

)

=

z

⁢

log

x

⁡

(

y

)

log

x

⁡

(

1

)

=

0

log

x

⁡

(

x

)

=

1

log

x

⁡

(

y

)

⁢

log

y

⁡

(

x

)

=

1

log

y

⁡

(

z

)

=

log

x

⁡

(

z

)

log

x

⁡

(

y

)

By the very first identity, any logarithm restricted (

http://planetmath.org/RestrictionOfAFunction

) to the set of positive integers is an

additive function

.

Notes.

In essence, logarithms convert

multiplication

to

addition

, and exponentiation to multiplication. Historically, these
properties of the logarithm made it a useful tool for doing numerical
calculations. Before the advent of electronic

calculators

and

computers

, tables of logarithms and the logarithmic slide rule were
essential computational aids.

Scientific applications predominantly make use of
logarithms whose base is the

Eulerian number

e

=

2.71828

⁢

…

.
Such logarithms are called

natural logarithms

and are commonly
denoted by the symbol

ln

, e.g.

ln

⁡

(

e

)

=

1

.

Natural logarithms naturally give rise to the natural logarithm

function

.

A frequent convention, seen in elementary mathematics texts and on
calculators, is that logarithms that do not give a base explicitly are
assumed to be base

10

, e.g.

log

⁡

(

100

)

=

2

.

This is far from . In Rudin’s “Real and

Complex analysis

”, for example, we see a baseless

log

used to
refer to the natural logarithm. By contrast, computer science and
information theory texts often assume 2 as the default logarithm base.
This is motivated by the fact that

log

2

⁡

(

N

)

is the approximate
number of bits required to encode

N

different messages.

The invention of logarithms is commonly credited to John Napier [

http://www-groups.dcs.st-and.ac.uk/ history/Mathematicians/Napier.html

Biography]

Title

logarithm

Canonical

name

Logarithm

Date of creation

2013-03-22 12:25:21

Last modified on

2013-03-22 12:25:21

Owner

rmilson (146)

Last modified by

rmilson (146)

Numerical id

21

Author

rmilson (146)

Entry type

Definition

Classification

msc 26A09

Classification

msc 26A06

Classification

msc 26-00

Related topic

Entropy

Related topic

ComplexLogarithm

Defines

base

Defines

natural logarithm

Defines

ln

Defines

log

Generated on Fri Feb 9 19:47:29 2018 by

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nth root formulas

nth root formulas

In the following formulas,

a

is a nonnegative

real number

and other letters

positive integers

.  For other formulas, see
the parent entry (

http://planetmath.org/NthRoot

).

1.

0

n

=

0

,

1

n

=

1

2.

a

1

=

a

3.

a

n

m

=

a

m

⁢

n

=

a

m

n

4.

a

m

⁢

k

n

⁢

k

=

a

m

n

5.

a

m

n

=

(

a

n

)

m

6.

a

m

⋅

a

n

=

a

m

+

n

m

⁢

n

Proof.

For proving, one uses the definition of

http://planetmath.org/node/754

n

th root and the power (

http://planetmath.org/GeneralAssociativity

) laws.

1

∘

.

0

n

=

0

,

1

n

=

1

2

∘

.

a

1

=

a

3

∘

.

(

a

n

m

)

m

⁢

n

=

(

(

a

n

m

)

m

)

n

=

(

a

n

)

n

=

a

4

∘

.

(

a

m

n

)

n

⁢

k

=

(

(

a

m

n

)

n

)

k

=

(

a

m

)

k

=

a

m

⁢

k

5

∘

.

(

(

a

n

)

m

)

n

=

(

(

a

n

)

n

)

m

=

a

m

6

∘

.

(

a

m

⋅

a

n

)

m

⁢

n

=

(

a

m

)

m

⁢

n

⁢

(

a

n

)

m

⁢

n

=

(

(

a

m

)

m

)

n

⁢

(

(

a

n

)

n

)

m

=

a

n

⁢

a

m

=

a

m

+

n

Title

nth root formulas

Canonical name

NthRootFormulas

Date of creation

2014-10-25 17:59:04

Last modified on

2014-10-25 17:59:04

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A09

Synonym

root formulas

Synonym

root formulae

Related topic

FractionPower

Generated on Fri Feb 9 19:47:32 2018 by

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rational sine and cosine

rational sine and cosine

Theorem

.

The only

acute angles

, whose sine and cosine are

rational

, are those determined by the

Pythagorean triplets

(

a

,

b

,

c

)

.

Proof.

1

o

¯

. When the

catheti

a

,

b

and the

hypotenuse

c

of a

right triangle

are

integers

, i.e. they form a Pythagorean triplet, then the sine

a

c

and the cosine

b

c

of one of the acute angles of the triangle are

rational numbers

.

2

o

¯

. Let the sine and the cosine of an acute angle

ω

be rational numbers

sin

⁡

ω

=

a

c

,

cos

⁡

ω

=

b

d

,

where the integers

a

,

b

,

c

,

d

satisfy

gcd

⁡

(

a

,

c

)

=

gcd

⁡

(

b

,

d

)

=

1

.

(1)

Since the square sum of sine and cosine is always 1, we have

a

2

c

2

+

b

2

d

2

=

1

.

(2)

By removing the

denominators

we get the

Diophantine equation

a

2

⁢

d

2

+

b

2

⁢

c

2

=

c

2

⁢

d

2

.

Since two of its terms are

divisible

by

c

2

, also the third term

a

2

⁢

d

2

is divisible by

c

2

.  But because by (1), the integers

a

2

and

c

2

are

coprime

, we must have

c

2

∣

d

2

(see the corollary of Bézout’s lemma).  Similarly, we also must have

d

2

∣

c

2

.  The last

divisibility

relations mean that

c

2

=

d

2

,  whence (2) may be written

a

2

+

b

2

=

c

2

,

and accordingly the sides

a

,

b

,

c

of a corresponding right triangle are integers.

Title

rational sine and cosine

Canonical

name

RationalSineAndCosine

Date of creation

2013-03-22 17:54:50

Last modified on

2013-03-22 17:54:50

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

8

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A09

Classification

msc 11D09

Classification

msc 11A67

Related topic

RationalPointsOnTwoDimensionalSphere

Related topic

GreatestCommonDivisor

Related topic

GeometricProofOfPythagoreanTriplet

Related topic

RationalBriggsianLogarithmsOfIntegers

Related topic

AlgebraicSinesAndCosines

Generated on Fri Feb 9 19:47:37 2018 by

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