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|