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If sin A + cos A = p and sin3A + cos3A = q, then which one of the following is correct?
- a)p3 – 3p + q = 0
- b)q3 – 3p + 2p = 0
- c)p3 – 3p + 2q = 0
- d)p3 + 3p + 2q = 0
Correct answer is option 'C'. Can you explain this answer?
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If sin A + cos A = p and sin3A+ cos3A = q, then which one of the follo...
(sin A + cos A)2 = sin2 A + cos2 A + 2 sin A cos A [∵ (a + b)2 = a2 + b2 + 2ab]
p2 = 1 + 2 sin A cosA [∵ sin2 A + cos2 A = 1]
∴ sin A cos A = (p2 – 1) /2
(sin A + cos A)3 = sin3 A + cos3 A + 3 sin A cos A (sin A + cos A)
[∵ (a + b)3 = a3 + b3 + 3ab (a + b)]
p3 = q + 3 × (p2 – 1)/2 × p
p3 = q + 3p3/2 – 3p/2
(0 = q + p3/2 -3p/2) × 2
0 = 2q + p3 – 3p
∴ p3 – 3p + 2q = 0
∴ sin A cos A = (p2 – 1) /2
(sin A + cos A)3 = sin3 A + cos3 A + 3 sin A cos A (sin A + cos A)
[∵ (a + b)3 = a3 + b3 + 3ab (a + b)]
p3 = q + 3 × (p2 – 1)/2 × p
p3 = q + 3p3/2 – 3p/2
(0 = q + p3/2 -3p/2) × 2
0 = 2q + p3 – 3p
∴ p3 – 3p + 2q = 0
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If sin A + cos A = p and sin3A+ cos3A = q, then which one of the follo...
We can use the triple angle formulas to express sin3A and cos3A in terms of sinA and cosA:
sin3A = 3sinA - 4sin^3A
cos3A = 4cos^3A - 3cosA
Substituting these expressions into the given equation sin3A * cos3A = q, we get:
(3sinA - 4sin^3A)(4cos^3A - 3cosA) = q
Expanding and simplifying this equation, we have:
12sinAcos^3A - 9sinAcosA - 16sin^3Acos^3A + 12sin^3AcosA = q
Factoring out sinAcosA from the first two terms and sin^3Acos^3A from the last two terms, we get:
sinAcosA(12cos^2A - 9 - 16sin^2Acos^2A + 12sin^2A) = q
Using the identity cos^2A + sin^2A = 1, we can simplify this equation further:
sinAcosA(12 - 9 - 16(1 - cos^2A)cos^2A + 12(1 - cos^2A)) = q
sinAcosA(12 - 9 - 16cos^2A + 16cos^4A + 12 - 12cos^2A) = q
sinAcosA(24 - 28cos^2A + 16cos^4A) = q
Since we know that sinAcosA = p, we can substitute p into the equation:
p(24 - 28cos^2A + 16cos^4A) = q
Now, we can compare this equation to the answer choices. If we let p^3 = p * p * p, the only answer choice that matches the equation is a) p^3.
sin3A = 3sinA - 4sin^3A
cos3A = 4cos^3A - 3cosA
Substituting these expressions into the given equation sin3A * cos3A = q, we get:
(3sinA - 4sin^3A)(4cos^3A - 3cosA) = q
Expanding and simplifying this equation, we have:
12sinAcos^3A - 9sinAcosA - 16sin^3Acos^3A + 12sin^3AcosA = q
Factoring out sinAcosA from the first two terms and sin^3Acos^3A from the last two terms, we get:
sinAcosA(12cos^2A - 9 - 16sin^2Acos^2A + 12sin^2A) = q
Using the identity cos^2A + sin^2A = 1, we can simplify this equation further:
sinAcosA(12 - 9 - 16(1 - cos^2A)cos^2A + 12(1 - cos^2A)) = q
sinAcosA(12 - 9 - 16cos^2A + 16cos^4A + 12 - 12cos^2A) = q
sinAcosA(24 - 28cos^2A + 16cos^4A) = q
Since we know that sinAcosA = p, we can substitute p into the equation:
p(24 - 28cos^2A + 16cos^4A) = q
Now, we can compare this equation to the answer choices. If we let p^3 = p * p * p, the only answer choice that matches the equation is a) p^3.
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If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer? for Defence 2025 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer?.
If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer? for Defence 2025 is part of Defence preparation. The Question and answers have been prepared according to the Defence exam syllabus. Information about If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Defence 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer?.
Solutions for If sin A + cos A = p and sin3A+ cos3A = q, then which one of the following is correct?a)p3– 3p + q = 0b)q3– 3p + 2p = 0c)p3– 3p + 2q = 0d)p3+ 3p + 2q = 0Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Defence.
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