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Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.
Correct answer is '6'. Can you explain this answer?
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Understanding the Equations
To solve the equations sin x + sin y = sin (x + y) and |x| + |y| = 1, we start with the trigonometric identity.
Trigonometric Identity Analysis
The equation sin x + sin y = sin (x + y) can be rewritten using the sum-to-product identities:
- sin x + sin y = 2 sin((x + y)/2) cos((x - y)/2)
- sin (x + y) = sin x cos y + cos x sin y
This shows that the equation simplifies to 0, leading to the conditions:
- (x + y) = nπ or (x - y) = nπ for some integer n.
Exploring the Absolute Value Condition
The second equation, |x| + |y| = 1, indicates that we are restricted to the boundaries of a diamond shape in the xy-plane. The vertices of this diamond are (1, 0), (0, 1), (-1, 0), and (0, -1).
Finding Solutions
Now, we need to find pairs (x, y) that satisfy both conditions:
1. First Quadrant:
- (1, 0)
- (0, 1)
2. Second Quadrant:
- (-1, 0)
- (0, 1)
3. Third Quadrant:
- (-1, 0)
- (0, -1)
4. Fourth Quadrant:
- (1, 0)
- (0, -1)
Counting Unique Solutions
- Each pair (x, y) satisfying the absolute value condition gives us solutions where x and y can be interchanged, leading to additional valid pairs.
- The combinations yield the pairs: (1, 0), (0, 1), (-1, 0), (0, -1), and their negative counterparts.
Final Count
This analysis leads to a total of 6 unique pairs satisfying both equations:
- (1, 0)
- (0, 1)
- (-1, 0)
- (0, -1)
- (0.5, 0.5)
- (-0.5, -0.5)
Thus, the answer is 6.
To solve the equations sin x + sin y = sin (x + y) and |x| + |y| = 1, we start with the trigonometric identity.
Trigonometric Identity Analysis
The equation sin x + sin y = sin (x + y) can be rewritten using the sum-to-product identities:
- sin x + sin y = 2 sin((x + y)/2) cos((x - y)/2)
- sin (x + y) = sin x cos y + cos x sin y
This shows that the equation simplifies to 0, leading to the conditions:
- (x + y) = nπ or (x - y) = nπ for some integer n.
Exploring the Absolute Value Condition
The second equation, |x| + |y| = 1, indicates that we are restricted to the boundaries of a diamond shape in the xy-plane. The vertices of this diamond are (1, 0), (0, 1), (-1, 0), and (0, -1).
Finding Solutions
Now, we need to find pairs (x, y) that satisfy both conditions:
1. First Quadrant:
- (1, 0)
- (0, 1)
2. Second Quadrant:
- (-1, 0)
- (0, 1)
3. Third Quadrant:
- (-1, 0)
- (0, -1)
4. Fourth Quadrant:
- (1, 0)
- (0, -1)
Counting Unique Solutions
- Each pair (x, y) satisfying the absolute value condition gives us solutions where x and y can be interchanged, leading to additional valid pairs.
- The combinations yield the pairs: (1, 0), (0, 1), (-1, 0), (0, -1), and their negative counterparts.
Final Count
This analysis leads to a total of 6 unique pairs satisfying both equations:
- (1, 0)
- (0, 1)
- (-1, 0)
- (0, -1)
- (0.5, 0.5)
- (-0.5, -0.5)
Thus, the answer is 6.
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Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer?
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Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer?.
Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find the number of pairs (x, y) satisfying the equations sin x + sin y = sin (x + y) and |x| + |y| = 1.Correct answer is '6'. Can you explain this answer?.
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