Explain what Var¯ahamihira meant in the first line of verse 2 when he said that “a fourth part of [the dhruva] is [the square of the Sine] of Aries.” What does this say about the value of the Sine of 30◦? Use this same result to justify the claim made in the second line of verse 2; what does it say the Sine of 60◦is equal to?
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Varahamihira was an ancient Indian astronomer and mathematician who contributed significantly to the field of astrology and astronomy. In his work, he often used geometric and trigonometric concepts to explain celestial phenomena. Show more…
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$\mathrm{A}=\frac{1}{2} \mathrm{x}^{2} \sin \theta$ $\frac{\mathrm{d} \mathrm{A}}{\mathrm{dt}}=\mathrm{x} \sin \theta \frac{\mathrm{dx}}{\mathrm{dt}}+\frac{\mathrm{x}^{2}}{2} \cos \theta \frac{\mathrm{d} \theta}{\mathrm{dt}}$ $\quad=12 \times \frac{1}{\sqrt{2}} \times \frac{1}{12}+\frac{144}{2} \times \frac{1}{\sqrt{2}} \times \frac{\pi}{180}$ $=2^{1 / 2}\left(\frac{1}{2}+\frac{\pi}{5}\right)$
This set of exercises will draw on the ideas presented in this section and your general math background. Before the widespread use of calculators, values of the sine and cosine for selected angles in the interval $\left[0^{\circ}, 45^{\circ}\right]$ were given in tables. Why was there no need to list the sine and cosine for angles in the interval $\left(45^{\circ}, 90^{\circ}\right) ?$
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