How does the trigonometry used by Aryabhata I differ from what had been developed by Ptolemy four centuries earlier?common indicate that the Hindus received their knowledge of trigonometry from the Greeks?

Mathematics by Geoffrey

Complete 15 of the following problems. Include all work and explanation needed to fully answer the question.

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1. What application motivates the mathematics included in the Sulva Sutras?

2. What mathematical subjects studied by Indian mathematicians long ago have no counterpart in the other cultures studied up to this point?

3. One reflection of Mesopotamian influence in India is the division of the circle into 360 degrees. Does having this system in

4. Use Aryabhata’s rule to compute the altitude of the sun above the horizon in London (latitude 5132’) at 10:00 AM (local solar time) on the vernal equinox. Assume that the sun rises at 6:00 AM on that day and sets at 6:00PM.

5. How does the trigonometry used by Aryabhata I differ from what had been developed by Ptolemy four centuries earlier?common indicate that the Hindus received their knowledge of trigonometry from the Greeks?

6. Besides the sine function, we also use the tangent and secant and their cofunctions. What is the origin of the words tangent and secant (in Latin), and why are they applied to the objects of trigonometry?

7. Given the Pell equation , which has solutions x = 3, y = 10 and x = 60, y = 199, construct a third solution and use it to get an approximation to .

8. Solve Bhaskara’s problem of finding the number of positive integers having five nonzero digits whose sum is 13.

9. How accurate are the rules given by Brahmagupta for computing areas and volumes?

10. How did Bhaskara II treat division by zero?

11. How is the Pythagorean theorem treated in the Zhou Bi Suan Jing?

12. Find all the solutions of the cubic equation without doing any numerical approximation. [Hint: If there is a rational solution r = m/n, then m must divide 243 and n must divide 2.]

13. Why were the Chinese mathematicians undeterred by the prospect of solving equations of degree 4 and higher?

14. Compare the use of thin slices of a solid figure for computing areas and volumes, as illustrated by Archimedes’ Method, Bhaskara’s computation of the area of a sphere, and Zu Chongzhi’s computation of the volume of a sphere. What differences among the three do you notice?

15. What areas of mathematics became specialties in Japan, and what innovations arose there?