test of mathematics at 10 +2 level

This is a Test of Mathematics Solution (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

Also see: Cheenta I.S.I. & C.M.I. Entrance Course

Problem

The value of the sum $$ \cos \frac{\pi}{1000} + \cos \frac{2\pi}{1000} + \cdots + \cos \frac {999 \pi}{1000} $$ is

(A) 0; (B)1; (C) \( \frac {1}{1000} \); (D) an irrational number;

Sequential Hints


(How to use this discussion: Do not read the entire solution at one go. First, read more on the Key Idea, then give the problem a try. Next, look into Step 1 and give it another try and so on.)

Key Idea

This is the generic use case of Complex Number \( \iota =\sqrt {-1} \) and De Moivre’s Theorem


Hint 1

We know that \( \cos (\pi – \theta) = – \cos \theta \). There are several ways to imagine this. One intuitive way is horizontal projection for \( \theta \) and \( \pi – \theta \) are of same magnitude but of opposite sides.

Now notice that if \( \theta = \frac{\pi}{1000} \) then \( \pi – \theta = \pi – \frac{ \pi}{1000} = \frac{999 \pi}{1000} \)

Hint 2

This implies \( \frac{\pi}{1000} + \frac{999 \pi}{1000} = 0 \). Similarly \( \frac{2\pi}{1000} + \frac{998 \pi}{1000} = 0 \) and so on.

Hint 3

Hence by properly pairing up, all of them will cancel out. Only \( \cos \frac{500\pi}{1000} = 0 \)