Multiplying Large Numbers Using Sine and Cosine

Date: 12/17/2002 at 14:31:58
From: Justin
Subject: Multiplication using sine and cosine

I am researching Tycho Brahe and have come upon an example where he 
uses [sin(a + b) + sin(a - b)]/2 to multiply large numbers together 
due to the availability of sine tables. The example given is:

"If a person wanted to find the product of a problem such as 155 x 
36, a person would find the angle values for the decimals of these 
numbers. Thus, sin 77°49' = 0.15500 and cos 68°54’ = 0.36000. Thus, 
36 x 155 = 105 [sin 77°49' + sin (-59°59')] / 2 = (97748 - 86588) / 2 
= 5580. Some significant figures would be lost but for large numbers 
this procedure saves labor."

I am unable to come up with the same results or figure out how this 
works. Most of the confusion arises with my unfamiliarity with degrees 
and minutes and how I would use them in a calculation. I have tried 
going backward from 0.15500, etc... to no avail. I am also not sure 
how this would be a universal algorithm.

Date: 12/17/2002 at 16:46:54
From: Doctor Peterson
Subject: Re: Multiplication using sine and cosine

Hi, Justin.

I found a page that describes this method in some detail:

   Henry Briggs - Introduction: the need for speed in calculation
   http://cerebro.cs.xu.edu/math/math147/02f/briggs/briggsintro.html 

The essential formula given there is

  cos(a)cos(b) = [cos(a + b) + cos(a - b)]/2

(Your version uses sines, but the work will be similar. You should be 
able to prove both formulas using the angle-sum identities.)

This, like logarithms, converts a multiplication on the left to an 
addition on the right. Since cosines are never greater than one, you 
have to move the decimal point in the given numbers while you do the 
work, and then insert the decimal point at the right place in the end, 
just as when you multiply decimals by hand. To multiply 155 by 36, 
we will be actually using the formula to multiply 0.155 by 0.36. We 
use the inverse cosine (that is, look up the angle whose cosine in the 
table is as close as possible to each of these numbers) to find

    cos(a) = 0.155 ==> a = 81.0832°
    cos(b) = 0.36  ==> b = 68.8998°

Note that you don't HAVE to use minutes and seconds if you don't want 
to; that is not an essential part of this process, but arose from the 
fact that their trig tables used them. I used my calculator and got 
decimal degrees; or I could use radians just as well.

Now what will the right side of the equation look like with these 
values?

    cos(a+b) = cos(149.9830°) = -0.865877
    cos(a-b) = cos(12.1834°) = 0.977477

Averaging these, we find that the product of 0.155 and 0.36 is

    (-0.865877 + 0.977477)/2 = 0.1116/2 = 0.0558

To convert this back to 155 * 36, we have to multiply by 1000*100, 
moving the decimal point right 5 places; so the answer is 5580. This 
is correct; we found it by a few table lookups, a few additions and 
subtractions, and a division by 2. Back in a time when people were 
afraid to depend on memorized multiplication tables, that was a big 
savings, especially for larger numbers.

Now, suppose you did have to use degrees and minutes. The only place 
where you have to do arithmetic on them is when you add and subtract 
the angles. We have

    a = cos^-1(0.155) = 81° 4' 59"
    b = cos^-1(0.36) = 68° 53' 59"

according to my calculator. To add them, just add the parts, first:

    a+b = (81+68)° (4+53)' (59+59)" = 149° 57' 118"

Everything is fine except the seconds, which should be less than 60; 
so we change 60 seconds to 1 minute and have

    118" = 60" + 58" = 1' 58"

so

    a+b = 149° 58' 58"

You do the same sort of thing to subtract:

    a-b = (81-68)° (4-53)' (59-59)"

We can't have a negative number of minutes, so we "borrow" a degree 
and add 60 minutes:

    a-b = (13-1)° (60+4-53)' 0" = 12° 11'

Now you take the cosine of these, so there's no more to do with 
minutes. All this carrying and borrowing should look familiar from 
elementary arithmetic; we're actually working in base 60 
(sexagesimal), a leftover from the ancient Babylonian number system!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 12/17/2002 at 18:34:53
From: Justin
Subject: Thank you (Multiplication using sine and cosine)

Thank you very much for you help. Your explanation was very thorough 
and will definitely help me put the finishing touches on my research 
paper.

Justin