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 |

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