This is a Test of Mathematics Solution Subjective 127 (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

Find all (x, y) such that sin x + sin y = sin (x+y) and |x| + |y| = 1

Discussion

|x| + |y| =1 is easier to plot. We have to treat the cases separately.

First quadrant: x +y = 1

Second quadrant: -x + y = 1 (since |x| = -x when x is negative)

Third Quadrant: -x-y =1

Fourth Quadrant: x – y =1

Now we work on sin x + sin y = sin (x + y).

This implies . Hence we have two possibilities:

OR

or

The above situations can happen when when

or or , where k is any integer.

Thus we need to plot the class of lines , and , and consider the intersection points of these lines with the graph of |x| + |y| = 1.

Clearly only for k=0, such intersection points can be found.