trigonometric formulas from series

trigonometric formulas from series

One may define the sine and the

cosine functions

for real (and

complex

)

arguments

using the

power series

sin

x

=

x

-

x

3

3

!

+

x

5

5

!

-

+

…

,

(1)

cos

x

=

1

-

x

2

2

!

+

x

4

4

!

-

+

…

,

(2)

and using only the properties of power series, easily derive most of the

goniometric formulas

, without any

geometry

.  For example, one gets instantly from (1) and (2) the values

sin

⁡

0

=

0

,

cos

⁡

0

=

1

and the parity relations (

http://

planetmath

.org/EvenoddFunction

)

sin

⁡

(

-

x

)

=

-

sin

⁡

x

,

cos

⁡

(

-

x

)

=

cos

⁡

x

.

Using the

Cauchy multiplication rule

for series one can obtain the

addition formulas

{

sin

⁡

(

x

+

y

)

=

sin

⁡

x

⁢

cos

⁡

y

+

cos

⁡

x

⁢

sin

⁡

y

,

cos

⁡

(

x

+

y

)

=

cos

⁡

x

⁢

cos

⁡

y

-

sin

⁡

x

⁢

sin

⁡

y

.

(3)

These produce straightforward many other important formulae, e.g.

sin

2

x

=

2

sin

x

cos

x

,

cos

2

x

=

cos

2

x

-

sin

2

x

(

y

=

:

x

)

(4)

and

cos

2

x

+

sin

2

x

=

1

(

y

=

:

-

x

)

.

(5)

The value

cos

⁡

π

2

=

0

,  as well as the formulae expressing the

periodicity

of sine and cosine, cannot be directly obtained from the series (1) and (2) — in fact, one must define the number

π

by using the

function

properties of the and its

derivative

series (

http://planetmath.org/PowerSeries

).

The equation

cos

⁡

x

=

0

has on the

interval

(

0

,

2

)

exactly one root (

http://planetmath.org/Equation

).  Actually, as sum of a power series,

cos

⁡

x

is

continuous

,

cos

⁡

0

=

1

>

0

and

cos

⁡

2

<

1

-

2

2

2

!

+

2

4

4

!

<

0

(see Leibniz’ estimate for

alternating series

(

http://planetmath.org/LeibnizEstimateForAlternatingSeries

)), whence there is at least one root.  If there were more than one root, then the derivative

-

sin

x

=

-

x

+

x

3

3

!

-

+

…

=

-

x

(

1

-

x

2

3

!

+

-

…

)

would have at least one zero on the interval; this is impossible, since by Leibniz the series in the parentheses does not change its sign on the interval:

1

-

x

2

3

!

+

-

…

>

1

-

2

2

3

!

>

0

Accordingly, we can define the number pi to be the least positive solution of the equation

cos

⁡

x

=

0

, multiplied by 2.

Thus we have

0

<

π

<

4

and

cos

⁡

π

2

=

0

.  Furthermore, by (5),

sin

⁡

π

2

=

1

,

and by (4),

sin

⁡

π

=

0

,

cos

⁡

π

=

-

1

,

sin

⁡

2

⁢

π

=

0

,

cos

⁡

2

⁢

π

=

1

.

Consequently, the addition formulas (3) yield the periodicities (

http://planetmath.org/PeriodicFunctions

)

sin

⁡

(

x

+

2

⁢

π

)

=

sin

⁡

x

,

cos

⁡

(

x

+

2

⁢

π

)

=

cos

⁡

x

.

Title

trigonometric formulas from series

Canonical name

TrigonometricFormulasFromSeries

Date of creation

2013-03-22 18:50:47

Last modified on

2013-03-22 18:50:47

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

9

Author

pahio (2872)

Entry type

Derivation

Classification

msc 26A09

Synonym

series definition of sine and cosine

Related topic

RigorousDefinitionOfTrigonometricFunctions

Related topic

ApplicationOfFundamentalTheoremOfIntegralCalculus

Related topic

TrigonometricFormulasFromDeMoivreIdentity

Related topic

GoniometricFormulae

Defines

π

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