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.

## 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 $$\cos \frac{500\pi}{1000} = 0$$