CAT Exam > CAT Questions > Given that cos x + cos y = 1, the range of si...
Start Learning for Free
Given that cos x + cos y = 1, the range of sin x - sin y is
- a)[-1, 1]
- b)[-2, 2]
- c)[0, √3]
- d)[-√3, √3]
Correct answer is option 'D'. Can you explain this answer?
Most Upvoted Answer
Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)...
Let’s say sin x – sin y = k. So we intend to find the range of k.
We have cos x + cos y = 1
So, k2 + 12 = (sin x - sin y )2 + (cos x + cosy )2
k2 + 12 = (sin x)2 + (sin y)2 – 2(sin x)(sin y) + (cos x)2 + (cos y)2 + 2(cos x)(cos y)
k2 + 12= sin2 x + cos2 x + sin2 y + cos2 y + 2(cos x)(cos y) – 2(sin x)(sin y)
k2 + 12 = 1 + 1 + 2(cos x)(cos y) – 2(sin x)(sin y) {Because, sin2 x + cos2 x = 1}
k2 + 12= 2 + 2[(cos x)(cos y) – (sin x)(sin y)] {Because, (cos x)(cos y) – (sin x)(sin y) = cos(x + y)}
k2 = 1 + 2(cos(x + y))
The maximum value that a cos function can take is 1, and the minimum value is -1.
Therefore the maximum value that k2 can take is 1+2(1) = 3,
and the minimum value that k2 can take is 1+2(-1) = -1,
This is impossible, k2 is always non-negative.
The minimum value that k2 can take is 0.
Therefore (k2)max = 3 and (k2)min = 0.
To get the range of k, we concentrate on (k2)max.
(k2)max = 3, implies that the extreme values of k are:
k = +√3 or -√3
So, the values of k range between +√3 and -√3, both +√3 and -√3 inclusive.
k = sin x – sin y, Remember!!
Therefore, Range of sin x – sin y = [-√3, √3]
We have cos x + cos y = 1
So, k2 + 12 = (sin x - sin y )2 + (cos x + cosy )2
k2 + 12 = (sin x)2 + (sin y)2 – 2(sin x)(sin y) + (cos x)2 + (cos y)2 + 2(cos x)(cos y)
k2 + 12= sin2 x + cos2 x + sin2 y + cos2 y + 2(cos x)(cos y) – 2(sin x)(sin y)
k2 + 12 = 1 + 1 + 2(cos x)(cos y) – 2(sin x)(sin y) {Because, sin2 x + cos2 x = 1}
k2 + 12= 2 + 2[(cos x)(cos y) – (sin x)(sin y)] {Because, (cos x)(cos y) – (sin x)(sin y) = cos(x + y)}
k2 = 1 + 2(cos(x + y))
The maximum value that a cos function can take is 1, and the minimum value is -1.
Therefore the maximum value that k2 can take is 1+2(1) = 3,
and the minimum value that k2 can take is 1+2(-1) = -1,
This is impossible, k2 is always non-negative.
The minimum value that k2 can take is 0.
Therefore (k2)max = 3 and (k2)min = 0.
To get the range of k, we concentrate on (k2)max.
(k2)max = 3, implies that the extreme values of k are:
k = +√3 or -√3
So, the values of k range between +√3 and -√3, both +√3 and -√3 inclusive.
k = sin x – sin y, Remember!!
Therefore, Range of sin x – sin y = [-√3, √3]
|
Explore Courses for CAT exam
|
|
Top Courses for CATView all
Question Description
Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer?.
Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? for CAT 2025 is part of CAT preparation. The Question and answers have been prepared according to the CAT exam syllabus. Information about Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? covers all topics & solutions for CAT 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer?.
Solutions for Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? in English & in Hindi are available as part of our courses for CAT.
Download more important topics, notes, lectures and mock test series for CAT Exam by signing up for free.
Here you can find the meaning of Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer?, a detailed solution for Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? has been provided alongside types of Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Given that cos x + cos y = 1, the range of sin x - sin y isa)[-1, 1]b)[-2, 2]c)[0, √3]d)[-√3, √3]Correct answer is option 'D'. Can you explain this answer? tests, examples and also practice CAT tests.
|
|
Explore Courses for CAT exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily