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If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of the equation
  • a)
    x2 – 6x + 1 = 0
  • b)
    6x2 – x – 1 = 0
  • c)
    6x2 + x + 1 = 0
  • d)
    x2 – x + 6 = 0
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of ...
Given Equation
To solve sin θ + 7 cos θ = 5, we want to express it in terms of tan(θ/2).
Using Trigonometric Identities
We can use the identities:
- sin θ = 2tan(θ/2) / (1 + tan²(θ/2))
- cos θ = (1 - tan²(θ/2)) / (1 + tan²(θ/2))
Let x = tan(θ/2).
Rewriting the Equation
Substituting the identities into the original equation gives:
2x / (1 + x²) + 7(1 - x²) / (1 + x²) = 5
Combining terms results in:
(2x + 7 - 7x²) / (1 + x²) = 5
Cross Multiplying
Cross-multiplying gives us:
2x + 7 - 7x² = 5(1 + x²)
This simplifies to:
2x + 7 - 7x² = 5 + 5x²
Rearranging Terms
Rearranging the equation yields:
-12x² + 2x + 2 = 0
To simplify, multiply by -1:
12x² - 2x - 2 = 0
Dividing by 2 gives:
6x² - x - 1 = 0
Conclusion
Thus, tan(θ/2) is a root of the equation 6x² - x - 1 = 0, confirming that option 'B' is correct.
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If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of the equationa)x2–6x + 1 = 0b)6x2–x –1 = 0c)6x2+ x + 1 = 0d)x2–x + 6 = 0Correct answer is option 'B'. Can you explain this answer?
Question Description
If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of the equationa)x2–6x + 1 = 0b)6x2–x –1 = 0c)6x2+ x + 1 = 0d)x2–x + 6 = 0Correct answer is option 'B'. 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 If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of the equationa)x2–6x + 1 = 0b)6x2–x –1 = 0c)6x2+ x + 1 = 0d)x2–x + 6 = 0Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin θ + 7 cos θ = 5, then tan (θ/2) is a root of the equationa)x2–6x + 1 = 0b)6x2–x –1 = 0c)6x2+ x + 1 = 0d)x2–x + 6 = 0Correct answer is option 'B'. Can you explain this answer?.
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