सिद्धान्तशिरोमणि: गणिताध्याय (भास्कराचार्य - ग्रहगणित, मध्यमाधिकार व स्पष्टाधिकार सटीक)

109 in the same commentary cited above. To start with, the H sines of 18°, 30°, 36°, 45°, 54°, 60° are known. They are respectively H₆, H₁₀, H₁₂, H₁₅, H₁₈, H₂₀. Also H₃₀ i.e. H sin 90° = R is also known. Formulae I and II will help us to derive, from H₆, H₃ and from H₃, H₂₇ ; also from H₆, we derive H₂₄. From H₁₀, H₅ and from H₅, H₂₅ and again from H₁₈, H₉ and from H₉, H₂₁ are derived. The remaining H sines sixteen in number cannot be got from either of the formulae. To meet the situation Bhāskara gives other formulae of his own discovery as he says “ प्रवक्ष्येऽथ विशिष्टमस्मात् ” i.e. “ I shall tell something more than this. These formulae he gives in the verses 12 to 15. They are H Sin ((90 ± x) / 2) = √((R² ± R H Sin x) / 2) III H Sin ((x − y) / 2) = √((H sin x + H sin y)² + (H cos x − H cos y)²) IV √(((H Cos x − H Sin x)²) / 2) = H Sin (45 − x) V R − (2 H Sin² x) / R = H Sin (90 − 2x) VI These formulae correspond to the modern formulae Sin (45 ± x/2)° = √((1 ± Sin x) / 2) Sin ((x − y) / 2) = √(((Sin x + Sin y)² + (Cos x − Cos y)²) / 2) √(((Cos x − Sin x)²) / 2) = Sin (45 − x) 1 − 2 Sin² x = Cos 2x respectively