Prosthaphaeresis formulas

Prosthaphaeresis formulas

The

Prosthaphaeresis formulas

convert sums of sines or cosines to products of them:

sin

⁡

A

+

sin

⁡

B

=

2

⁢

sin

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

sin

⁡

A

-

sin

⁡

B

=

2

⁢

sin

⁡

(

A

-

B

2

)

⁢

cos

⁡

(

A

+

B

2

)

cos

⁡

A

+

cos

⁡

B

=

2

⁢

cos

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

cos

⁡

A

-

cos

⁡

B

=

-

2

⁢

sin

⁡

(

A

+

B

2

)

⁢

sin

⁡

(

A

-

B

2

)

We prove the first two using the sine of a sum and sine of a

difference

formulas:

sin

⁡

(

X

+

Y

)

=

sin

⁡

X

⁢

cos

⁡

Y

+

cos

⁡

X

⁢

sin

⁡

Y

sin

⁡

(

X

-

Y

)

=

sin

⁡

X

⁢

cos

⁡

Y

-

cos

⁡

X

⁢

sin

⁡

Y

Adding or subtracting the two equations yields

sin

⁡

(

X

+

Y

)

+

sin

⁡

(

X

-

Y

)

=

2

⁢

sin

⁡

X

⁢

cos

⁡

Y

sin

⁡

(

X

+

Y

)

-

sin

⁡

(

X

-

Y

)

=

2

⁢

sin

⁡

Y

⁢

cos

⁡

X

If we let

X

=

A

+

B

2

and

Y

=

A

-

B

2

, then

X

+

Y

=

2

⁢

A

2

=

A

and

X

-

Y

=

2

⁢

B

2

=

B

, and the last two equations become

sin

⁡

A

+

sin

⁡

B

=

2

⁢

sin

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

sin

⁡

A

-

sin

⁡

B

=

2

⁢

sin

⁡

(

A

-

B

2

)

⁢

cos

⁡

(

A

+

B

2

)

as desired.

The last two can be proven similarly, this time using the cosine of a sum and cosine of a difference formulas:

cos

⁡

(

X

+

Y

)

=

cos

⁡

X

⁢

cos

⁡

Y

-

sin

⁡

X

⁢

sin

⁡

Y

cos

⁡

(

X

-

Y

)

=

cos

⁡

X

⁢

cos

⁡

Y

+

sin

⁡

X

⁢

sin

⁡

Y

Adding or subtracting the two equations yields

cos

⁡

(

X

+

Y

)

+

cos

⁡

(

X

-

Y

)

=

2

⁢

cos

⁡

X

⁢

cos

⁡

Y

cos

⁡

(

X

+

Y

)

-

cos

⁡

(

X

-

Y

)

=

-

2

⁢

sin

⁡

Y

⁢

sin

⁡

X

Again, if we let

X

=

A

+

B

2

and

Y

=

A

-

B

2

, then

X

+

Y

=

2

⁢

A

2

=

A

and

X

-

Y

=

2

⁢

B

2

=

B

, and the last two equations become

cos

⁡

A

+

cos

⁡

B

=

2

⁢

cos

⁡

(

A

+

B

2

)

⁢

cos

⁡

(

A

-

B

2

)

cos

⁡

A

-

cos

⁡

B

=

-

2

⁢

sin

⁡

(

A

-

B

2

)

⁢

sin

⁡

(

A

+

B

2

)

as desired.

Notes

’Prosthaphaeresis’ comes from the Greek: “prosthesi” =

addition

+ “afairo” =

subtraction

.

The Prosthaphaeresis formula

cos

⁡

x

⁢

cos

⁡

y

=

cos

⁡

(

x

+

y

)

+

cos

⁡

(

x

-

y

)

2

was used by scientists to transform

multiplication

into addition. For example, to calculate the product

a

⁢

b

, where

0

<

a

,

b

<

1

(for

a

and

b

outside of this range, it is a simple matter to multiply or divide by a factor of 10 and divide or multiply this back in later), one would let

cos

⁡

x

=

a

and

cos

⁡

y

=

b

. Using a table of cosines, one could then find an approximate value for

x

and

y

, then find

x

+

y

and

x

-

y

, and look up the cosines of the resulting two quantities (that is,

cos

⁡

(

x

+

y

)

and

cos

⁡

(

x

-

y

)

). The

average

of these numbers is the desired product

a

⁢

b

. This technique was used by Tycho Brahe to perform astronomical calculations.

Title

Prosthaphaeresis formulas

Canonical

name

ProsthaphaeresisFormulas

Date of creation

2013-03-22 14:33:55

Last modified on

2013-03-22 14:33:55

Owner

mathfanatic (5028)

Last modified by

mathfanatic (5028)

Numerical id

7

Author

mathfanatic (5028)

Entry type

Proof

Classification

msc 26A09

Synonym

Simpson’s formulas

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