Plane Trigonometry

Ratios of Certain Angles

690 sin45°=cos=12, tan45°=1 691 sin60°=32, cos60°=12, tan60°=3 692 sin15°=3−122, cos15°=3+122 tan15°=2−3 cot15°=2+3 } 693 sin18°=5−14, cos18°=5+522, tan18°=5−255 694 sin54°=5+14, cos54°=5−522, tan54°=5+255 695 By taking the complements of these angles, the same table gives the ratios of 30°, 75°, 72°, and 36°. 696

Proofs

sin15° is obtained from sin(45°−30°), expanded by (628). 697 sin18° from the equation sin2𝑥=cos3𝑥, where 𝑥=18°. 698 sin54° from sin3𝑥=3sin𝑥−4sin³𝑥, where 𝑥=18°. 699 And the ratios of various angles may be obtained by taking the sum, difference, or some multiple of the angles in the table, and making use of known formula. Thus 12°=30°−18°, 712°=15°2, ⋯

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive