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.
Find all (x, y) such that sin x + sin y = sin (x+y) and |x| + |y| = 1
|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:
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.
Hence required points are (0,1), (0,-1), (1,0), (-1,0), (1/2, -1/2), (-1/2, 1/2).
- 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)