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