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Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational
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Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a...
Proof that 'a' is irrational:
- Assume a is rational:
If 'a' is rational, then it can be expressed as a fraction a = p/q, where p and q are integers with q ≠ 0.
- Substitute a = p/q into the equation:
1 + sin^2 (p/q)x = cos x
- Rearrange the equation:
sin^2 (p/q)x = cos x - 1
sin^2 (p/q)x = -sin^2 x
- Apply the double-angle identity for sine:
sin^2 (p/q)x = 1 - cos(2x)
- Substitute x = π/4 into the equation:
sin^2 (p/q)(π/4) = 1 - cos(π/2)
sin^2 (p/4q)(π/4) = 1
- Since sin^2 (π/4) = 1/2:
(p/4q)(π/4) = 1
- Therefore:
p/q = 4
- This contradicts the assumption that a is irrational:
Since the assumption that 'a' is rational led to a contradiction, 'a' must be irrational.
Therefore, if the equation 1 + sin^2 ax = cos x has a unique solution, then 'a' must be irrational.
- Assume a is rational:
If 'a' is rational, then it can be expressed as a fraction a = p/q, where p and q are integers with q ≠ 0.
- Substitute a = p/q into the equation:
1 + sin^2 (p/q)x = cos x
- Rearrange the equation:
sin^2 (p/q)x = cos x - 1
sin^2 (p/q)x = -sin^2 x
- Apply the double-angle identity for sine:
sin^2 (p/q)x = 1 - cos(2x)
- Substitute x = π/4 into the equation:
sin^2 (p/q)(π/4) = 1 - cos(π/2)
sin^2 (p/4q)(π/4) = 1
- Since sin^2 (π/4) = 1/2:
(p/4q)(π/4) = 1
- Therefore:
p/q = 4
- This contradicts the assumption that a is irrational:
Since the assumption that 'a' is rational led to a contradiction, 'a' must be irrational.
Therefore, if the equation 1 + sin^2 ax = cos x has a unique solution, then 'a' must be irrational.
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Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational
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Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational.
Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Equation 1+ sin^2 ax = cos x has a unique solution then prove that 'a' is irrational.
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