Sin and Cos in Surds (original) (raw)

Julian D. A. Wiseman

Contents: values of Sin[α] and Cos[α], expressed in surds, for α=_n_×3° or α=_n_×5⅝°, _n_∈ℕ: Sin[0°], Sin[3°], Sin[5.625°], Sin[6°], Sin[9°], Sin[11.25°], Sin[12°], Sin[15°], Sin[16.875°], Sin[18°], Sin[21°], Sin[22.5°], Sin[24°], Sin[27°], Sin[28.125°], Sin[30°], Sin[33°], Sin[33.75°], Sin[36°], Sin[39°], Sin[39.375°], Sin[42°], Sin[45°], Sin[48°], Sin[50.625°], Sin[51°], Sin[54°], Sin[56.25°], Sin[57°], Sin[60°], Sin[61.875°], Sin[63°], Sin[66°], Sin[67.5°], Sin[69°], Sin[72°], Sin[73.125°], Sin[75°], Sin[78°], Sin[78.75°], Sin[81°], Sin[84°], Sin[84.375°], Sin[87°], and Sin[90°].

The table shows Sin[] and Cos[] in surds, for angles that are integer multiples of 3° or of 5⅝° = 90°/16. The surds are shown in several formats.

Help! Sin and Cos of 3°, 21°, 33°, 39°, 51°, 57°, 69°, and 87° each have two answers. One is longer, but the Sqrt’s are nested only two deep; one is more concise but the Sqrt’s are three deep. A concise two-deep answer would be preferred, and other simplifications are also welcomed. Credit will be given.

Errors: whilst the outputs have been tested, it is possible that errors remain. Please do test things before embedding them somewhere important—and if errors or possible improvements are found, tell the author.

Credit: in this post (on a password-protected bulletin board) Arthur L. Rubin suggested a method of simplifying Sin[6°]. The method was general and quite obvious, but not quite so obvious that the author thought of it first.

Julian D. A. Wiseman, June 2008