It could be the world's oldest relation between the sum and product of trigonometric function.

In the 16th century Europe, when decimal point has yet to be developed, Viete(1540-1603) tried to examine trigonometric function analytically.




                
                
                
                
In this equation, let , then, we get:

Thereafter, this equation was widely used in Europe as a method to simplify multiplication in the field of astronomical calculations.

For instance, say is 10,000, and prepare respective angular representation tables for the case of changing from 1 to 10,000, and changing from 1 to 10,000 as well. For angular measures, a sexagesimal system of angular measure (Examples: 98°25´40´´) must have been used, but we shall employ a sexagesimal system of angular measure inclusive of decimal point(Example: 98´.427777). In Europe those days, tables up to 12 digits and 15 digits existed.

Let's calculate as an example.
From the table, let's find angle that makes equal to 4852 and angle that makes equal to 6235. Then, calculate as below:

Look for in the table. If there no exact values are found, and then use those values that are near.

This gives us the value of:

Soon after this, because both Napier (1550-1617) and Briggs (1561-1630) invented common logarithm applicable to multiplication and division, this method was ceased to be used.

Corresponding Table
sin angle cos angle
1 0.005729578 1 89.99427042
2 0.011459156 2 89.98854084
3 0.017188734 3 89.98281127
4 0.022918312 4 89.97708169
5 0.028647891 5 89.97135211
6 0.03437747 6 89.96562253
7 0.040107049 7 89.95989295
8 0.045836628 8 89.95416337
9 0.051566209 9 89.94843379
10 0.057295789 10 89.94270421
11 0.06302537 11 89.93697463
12 0.068754952 12 89.93124505
13 0.074484534 13 89.92551547
14 0.080214118 14 89.91978588
4852 29.02557669
6235 51.42782406
9990 87.43744127 9990 2.562558733
9991 87.5689636 9991 2.431036405
9992 87.708016 9992 2.291983997
9993 87.85606316 9993 2.143936842
9994 88.01511672 9994 1.984883276
9995 88.18807286 9995 1.811927138
9996 88.37937661 9996 1.620623393
9997 88.59651067 9997 1.403489331
9998 88.85406531 9998 1.14593469
9999 89.18970856 9999 0.810291437


Japanese sites
Photograph| Mathematical Formulas| Kodawari House| Pinpoint StreetView|