trigonometric formulas from de Moivre identity

trigonometric formulas from de Moivre identity

De Moivre identity

(

cos

φ

+

i

sin

φ

)

n

=

cos

n

φ

+

i

sin

n

φ

(

n

∈

ℤ

)

(1)

implies simply some important trigonometric formulas, the

derivation

of which without

imaginary numbers

would require much longer calculations.

When one expands the left hand side of (1) using the binomial theorem (

n

>

0

), the sum of the real terms (the

real part

) must be

cos

⁡

n

⁢

φ

and the sum of the imaginary terms (cf. the

imaginary part

) must equal

i

⁢

sin

⁡

n

⁢

φ

.  Thus both

cos

⁡

n

⁢

φ

and

sin

⁡

n

⁢

φ

has been expressed as

polynomials

of

sin

⁡

φ

and

cos

⁡

φ

with integer coefficients.

For example, if

n

=

5

,  we have

(

cos

⁡

φ

+

i

⁢

sin

⁡

φ

)

5

=

cos

5

⁡

φ

+

5

⁢

i

⁢

cos

4

⁡

φ

⁢

sin

⁡

φ

-

10

⁢

cos

3

⁡

φ

⁢

sin

2

⁡

φ

-

10

⁢

i

⁢

cos

2

⁡

φ

⁢

sin

3

⁡

φ

+

5

⁢

cos

⁡

φ

⁢

sin

4

⁡

φ

+

i

⁢

sin

5

⁡

φ

,

whence

cos

⁡

5

⁢

φ

=

cos

5

⁡

φ

-

10

⁢

cos

3

⁡

φ

⁢

sin

2

⁡

φ

+

5

⁢

cos

⁡

φ

⁢

sin

4

⁡

φ

,

sin

⁡

5

⁢

φ

=

5

⁢

cos

4

⁡

φ

⁢

sin

⁡

φ

-

10

⁢

cos

2

⁡

φ

⁢

sin

3

⁡

φ

+

sin

5

⁡

φ

.

By the “fundamental formula”

sin

2

⁡

φ

+

cos

2

⁡

φ

=

1

of

trigonometry

, the even powers on the right hand sides may be expressed with the other

function

; therefore we obtain

cos

⁡

5

⁢

φ

=

16

⁢

cos

5

⁡

φ

-

20

⁢

cos

3

⁡

φ

+

5

⁢

cos

⁡

φ

,

(2)

sin

⁡

5

⁢

φ

=

16

⁢

sin

5

⁡

φ

-

20

⁢

sin

3

⁡

φ

+

5

⁢

sin

⁡

φ

.

(3)

0.1

Linearisation formulas

There are also

inverse

formulas where one expresses the integer powers

cos

m

⁡

φ

and

sin

n

⁡

φ

and their products as the polynomials with

rational

coefficients of either

cos

⁡

φ

,

cos

⁡

2

⁢

φ

, …  or

sin

⁡

φ

,

sin

⁡

2

⁢

φ

, …,  depending on whether it is a question of an even (

http://planetmath.org/EvenFunction

) or an

odd function

of

φ

.  We will derive the transformation formulas.

If we denote

cos

⁡

φ

+

i

⁢

sin

⁡

φ

:=

t

,

then the

complex conjugate

of

t

is the same as its inverse number:

cos

⁡

φ

-

i

⁢

sin

⁡

φ

=

1

t

.

By adding and subtracting, these equations yield

cos

⁡

φ

=

1

2

⁢

(

t

+

1

t

)

,

sin

⁡

φ

=

1

2

⁢

i

⁢

(

t

-

1

t

)

.

(4)

Similarly, the equations

(

cos

⁡

φ

+

i

⁢

sin

⁡

φ

)

±

n

=

cos

⁡

(

±

n

⁢

φ

)

+

i

⁢

sin

⁡

(

±

n

⁢

φ

)

yield

cos

⁡

n

⁢

φ

=

1

2

⁢

(

t

n

+

1

t

n

)

,

sin

⁡

n

⁢

φ

=

1

2

⁢

i

⁢

(

t

n

-

1

t

n

)

.

(5)

for any integer

n

.  The linearisation formulas are obtained by expanding first the expression to be linearised with the equations (4) and then simplifying the result with the equations (5).

Example 1.

cos

4

⁡

φ

=

(

1

2

⁢

(

t

+

1

t

)

)

4

=

1

16

⁢

(

t

4

+

4

⁢

t

2

+

6

+

4

t

2

+

1

t

4

)

=

1

16

⁢

(

t

4

+

1

t

4

)

+

1

4

⁢

(

t

2

+

1

t

2

)

+

3

8

=

1

8

⁢

cos

⁡

4

⁢

φ

+

1

2

⁢

cos

⁡

2

⁢

φ

+

3

8

Example 2.

cos

4

⁡

φ

⁢

sin

3

⁡

φ

=

1

16

⁢

(

t

+

1

t

)

4

⁢

-

1

8

⁢

i

⁢

(

t

-

1

t

)

3

=

-

1

128

⁢

i

⁢

(

t

2

-

1

t

2

)

3

⁢

(

t

+

1

t

)

=

-

1

128

⁢

i

⁢

(

t

6

-

3

⁢

t

2

+

3

t

2

-

1

t

6

)

⁢

(

t

+

1

t

)

=

-

1

128

⁢

i

⁢

(

t

7

-

3

⁢

t

3

+

3

t

-

1

t

5

-

3

⁢

t

+

3

t

3

-

1

t

7

)

=

-

1

128

⁢

i

⁢

(

(

t

7

-

1

t

7

)

+

(

t

5

-

1

t

5

)

-

3

⁢

(

t

3

-

1

t

3

)

-

3

⁢

(

t

-

1

t

)

)

=

-

1

64

⁢

sin

⁡

7

⁢

φ

-

1

64

⁢

sin

⁡

5

⁢

φ

+

3

64

⁢

sin

⁡

3

⁢

φ

+

3

64

⁢

sin

⁡

φ

Title

trigonometric formulas from de Moivre identity

Canonical name

TrigonometricFormulasFromDeMoivreIdentity

Date of creation

2013-03-22 18:51:16

Last modified on

2013-03-22 18:51:16

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

16

Author

pahio (2872)

Entry type

Derivation

Classification

msc 30D05

Classification

msc 30A99

Related topic

TrigonometricFormulasFromSeries

Related topic

ReductionFormulas

Related topic

GoniometricFormulae

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