Test of Mathematics Solution Subjective 127 -Graphing relations

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 $ \displaystyle{2 \sin \left ( \frac{x+y} {2} \right ) \cos \left ( \frac{x-y} {2} \right ) = 2 \sin \left ( \frac{x+y} {2} \right ) \cos \left ( \frac{x+y} {2} \right ) }$. Hence we have two possibilities:

$ \displaystyle{\sin \left ( \frac{x+y} {2} \right ) = 0 }$ OR
$ \displaystyle{\cos \left ( \frac{x+y} {2} \right ) - \cos \left ( \frac{x-y} {2} \right ) = 0 }$ or $ \displaystyle { \sin \frac{x}{2} \sin \frac{y}{2} } = 0 $

The above situations can happen when when

$ \displaystyle{ \frac{x +y}{2} = k \pi } $ or $ \displaystyle{\frac {x}{2} = k \pi }$ or $ \displaystyle{ \frac{y}{2} = k \pi }$, where k is any integer.

Thus we need to plot the class of lines $ \displaystyle{ x + y = 2 k \pi } $, $ \displaystyle{ x = 2k\pi } $ and $ \displaystyle{ y = 2k\pi } $, 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.

Hence required points are (0,1), (0,-1), (1,0), (-1,0), (1/2, -1/2), (-1/2, 1/2).

Chatuspathi

What is this topic: Graphing
What are some of the associated concept: Trigonometric Identities
Where can learn these topics: I.S.I. & C.M.I. Entrance Course of Cheenta, discusses these topics in the ‘Calculus’ module.
Book Suggestions: Play With Graphs (Arihant Publication)