Can these questions be solved above?

In this question, we will talk about the ancient way to compute 7. As * is defined as the area of a circle with radius 1, ancient people tried to approximate 7 with inscribed equilateral polygons. For example, an inscribed equilateral triangle of the unit circle has area 318. In general, we can denote S(N) as the area of an inscribed equilateral N-gon in the unit circle. (a) Prove that S(N) = > . sin( ) and SN converges 7. The convergence of S(N) to * is somehow more important than the way to compute them. The ancient way to prove S(N)'s convergence, however, does not use the prop- erties of trigonometric functions. Instead, an important inequality coming from geometric observations, can also help us prove the convergence. (b) Prove the Liu Hui's inequality: for every N 2 3, S(2N) ST S 2S(2N) - S(N). (Hint: Think geometrically, what does S(2N) - S(N) mean?) (c) Use Liu Hui's inequality to show lim S(3 . 2* ) = 1. K -+ DO (Hint: How does 7 - S(3 . 2*) evolve?) Now let's compute * ~ S(3 . 2*) using an iterative method. (d) Use the result in (a) to prove that for N 2 3, S(2N) 1 - (N/2 N 2 (e) We can quickly verify that S(3) = 3:3 by simple geometry. Use this as the initial data and compute S(3072) and S(24576). Here, 3072 = 3 . 210 and 24576 = 3 . 213. (Hint: Don't use manpower. Design an algorithm and use your preferred programming language.) (f) Briefly explain why ancient people didn't directly compute - sin(2" ) for large N, but preferred the iterative way. There're in fact multiple reasons, and you just need to come up with one reason