rigorous definition of trigonometric functions

rigorous definition of trigonometric functions

It is possible to define the

trigonometric functions

rigorously by means of a
process based upon the angle

addition

identities. A sketch of how this is done
is provided below.

To begin, define a

sequence

{

c

n

}

n

=

1

∞

by the

initial condition

c

1

=

1

and the recursion

c

n

+

1

=

1

-

1

-

c

n

2

.

Likewise define a sequence

{

s

n

}

n

=

1

∞

by the conditions

s

1

=

1

and

s

n

+

1

=

c

n

2

.

(In both equations above, we take the

positive

square root

.)
It may be shown that both of these sequences are

strictly decreasing

and approach

0

.

Next, define a sequence of

2

×

2

matrices as follows:

m

n

=

(

1

-

c

n

s

n

-

s

n

1

-

c

n

)

Using the recursion relations which define

c

n

and

s

n

, it may be shown that

m

n

+

1

2

=

m

n

, More grenerally, using induction, this can be generalised to

m

n

+

k

2

k

=

m

n

.

It is easy to check that the product of any two matrices of the form

(

x

y

-

y

x

)

is of the same form. Hence, for any

integers

k

and

n

, the matrix

m

n

k

will be of this form. We can therefore define

functions

S

and

C

from

rational numbers

whose

denominator

is a

power of two

to

real
numbers

by the following equation:

(

C

⁢

(

n

2

k

)

S

⁢

(

n

2

k

)

-

S

⁢

(

n

2

k

)

C

⁢

(

n

2

k

)

)

=

(

1

-

c

k

s

k

-

s

k

1

-

c

k

)

n

.

From the recursion relations, we may prove the following identities:

S

2

⁢

(

r

)

+

C

2

⁢

(

r

)

=

1

S

⁢

(

p

+

q

)

=

S

⁢

(

p

)

⁢

C

⁢

(

q

)

+

S

⁢

(

q

)

⁢

C

⁢

(

p

)

C

⁢

(

p

+

q

)

=

C

⁢

(

p

)

⁢

C

⁢

(

q

)

-

S

⁢

(

p

)

⁢

S

⁢

(

q

)

From the fact that

c

n

→

0

and

s

n

→

0

as

n

→

∞

, it
follows that, if

{

p

n

}

n

=

1

∞

and

{

q

n

}

n

=

1

∞

are
two sequences of rational numbers whose denominators are powers of two
such that

lim

n

→

∞

⁡

p

n

=

lim

n

→

∞

⁡

q

n

, then

lim

n

→

∞

⁡

C

⁢

(

p

n

)

=

lim

n

→

∞

⁡

C

⁢

(

q

n

)

and

lim

n

→

∞

⁡

S

⁢

(

p

n

)

=

lim

n

→

∞

⁡

S

⁢

(

q

n

)

. Therefore,
we may define functions by the conditions that, for any

convergent
series

of rational numbers

{

r

n

}

n

=

0

∞

whose denominators
are powers of two,

cos

⁡

(

π

⁢

lim

n

→

∞

⁡

r

n

)

=

lim

n

→

∞

⁡

C

⁢

(

r

n

)

and

sin

⁡

(

π

⁢

lim

n

→

∞

⁡

r

n

)

=

lim

n

→

∞

⁡

S

⁢

(

r

n

)

.

By

continuity

, we see that these functions satisfy the angle addition
identities.

Application

. Let us use the definitions above to find

sin

⁡

(

π

2

)

and

cos

⁡

(

π

2

)

. Let

r

i

:=

1

2

for every

positive integer

i

. Then we need to find

C

⁢

(

1

2

)

and

S

⁢

(

1

2

)

. We use the matrix above defining

C

and

S

, and set

n

=

k

=

1

:

(

C

⁢

(

1

2

)

S

⁢

(

1

2

)

-

S

⁢

(

1

2

)

C

⁢

(

1

2

)

)

=

(

1

-

c

1

s

1

-

s

1

1

-

c

1

)

=

(

0

1

-

1

0

)

.

As a result,

cos

⁡

(

π

2

)

=

cos

⁡

(

π

⁢

lim

i

→

∞

⁡

1

2

)

=

lim

i

→

∞

⁡

C

⁢

(

1

2

)

=

C

⁢

(

1

2

)

=

0

. Similarly,

sin

⁡

(

π

2

)

=

1

.

Title

rigorous definition of trigonometric functions

Canonical name

RigorousDefinitionOfTrigonometricFunctions

Date of creation

2013-03-22 16:22:11

Last modified on

2013-03-22 16:22:11

Owner

CWoo (3771)

Last modified by

CWoo (3771)

Numerical id

9

Author

CWoo (3771)

Entry type

Derivation

Classification

msc 26A09

Related topic

TrigonometricFormulasFromSeries

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