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If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relation
- a)a2 + b2 + 2ac = 0
- b)a2 – b2 + 2ac = 0
- c)a2 + c2 + 2ab = 0
- d)a2 – b2 – 2ac = 0
Correct answer is option 'B'. Can you explain this answer?
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If sinθ and cosθ are the roots of the equation ax2 –...
sinθ + cosθ = b/a
sinθ . cosθ = c/a
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If sinθ and cosθ are the roots of the equation ax2 –...
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If sinθ and cosθ are the roots of the equation ax2 –...
Understanding the Roots
If sin(θ) and cos(θ) are the roots of the quadratic equation ax² - bx + c = 0, we can use Vieta's formulas to find relationships between the coefficients a, b, and c.
Vieta's Formulas
- The sum of the roots (sin(θ) + cos(θ)) is given by b/a.
- The product of the roots (sin(θ) * cos(θ)) is given by c/a.
Using Trigonometric Identities
We know from trigonometric identities that:
- (sin(θ) + cos(θ))² = sin²(θ) + cos²(θ) + 2sin(θ)cos(θ)
- Since sin²(θ) + cos²(θ) = 1, we can express this as:
1 = (sin(θ) + cos(θ))² - 2sin(θ)cos(θ)
By substituting the Vieta's formulas, we have:
b²/a² - 2c/a = 1
Multiplying through by a² gives:
b² - 2ac = a²
Rearranging this leads to:
a² - b² + 2ac = 0
This matches option b), confirming that the coefficients a, b, and c indeed satisfy this relationship.
Conclusion
The correct relation derived from the roots sin(θ) and cos(θ) in the quadratic equation is:
a² - b² + 2ac = 0
This clearly shows how the coefficients are interrelated through the properties of sine and cosine.
If sin(θ) and cos(θ) are the roots of the quadratic equation ax² - bx + c = 0, we can use Vieta's formulas to find relationships between the coefficients a, b, and c.
Vieta's Formulas
- The sum of the roots (sin(θ) + cos(θ)) is given by b/a.
- The product of the roots (sin(θ) * cos(θ)) is given by c/a.
Using Trigonometric Identities
We know from trigonometric identities that:
- (sin(θ) + cos(θ))² = sin²(θ) + cos²(θ) + 2sin(θ)cos(θ)
- Since sin²(θ) + cos²(θ) = 1, we can express this as:
1 = (sin(θ) + cos(θ))² - 2sin(θ)cos(θ)
By substituting the Vieta's formulas, we have:
b²/a² - 2c/a = 1
Multiplying through by a² gives:
b² - 2ac = a²
Rearranging this leads to:
a² - b² + 2ac = 0
This matches option b), confirming that the coefficients a, b, and c indeed satisfy this relationship.
Conclusion
The correct relation derived from the roots sin(θ) and cos(θ) in the quadratic equation is:
a² - b² + 2ac = 0
This clearly shows how the coefficients are interrelated through the properties of sine and cosine.
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If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer?
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If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer? for 2025 is part of preparation. The Question and answers have been prepared according to the exam syllabus. Information about If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer?.
If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer? for 2025 is part of preparation. The Question and answers have been prepared according to the exam syllabus. Information about If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer?.
Solutions for If sinθ and cosθ are the roots of the equation ax2 – bx + c = 0, then a, b and c satisfy the relationa)a2 + b2 + 2ac = 0b)a2– b2+ 2ac = 0c)a2 + c2 + 2ab = 0d)a2 – b2 – 2ac = 0Correct answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for .
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