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In tanα = √a, where a is a rational number which is not a perfect square, then which of the following is a rational number?
- a)sin2α
- b)tan2α
- c)cos2α
- d)none
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
In tanα= √a, where a is a rational number which is not a p...
Understanding the Problem
Given the equation tan(α) = √a, where 'a' is a rational number that is not a perfect square, we need to explore which of the trigonometric values (sin²α, tan²α, cos²α) is a rational number.
Key Relationships
- Trigonometric Identities:
- tan²α = sin²α / cos²α
- sin²α + cos²α = 1
Analysis of tan²α
- Calculate tan²α:
- tan²α = a (since tan(α) = √a)
- Since 'a' is rational, tan²α is rational.
Analysis of sin²α
- Using the identity:
- sin²α = tan²α * cos²α
- We know tan²α is rational, but cos²α must be evaluated to determine sin²α's rationality.
Analysis of cos²α
- Using the sine and cosine relationship:
- cos²α = 1 / (1 + tan²α)
- cos²α = 1 / (1 + a)
- Since 'a' is rational, 1 + a is also rational.
- Therefore, cos²α = 1 / (1 + a) is a rational number.
Conclusion
- Final Evaluation:
- sin²α depends on the value of cos²α, which is rational.
- tan²α is rational.
- cos²α is rational.
Thus, the correct answer is option 'C', as cos²α is the only confirmed rational number when tan(α) = √a and 'a' is a non-perfect square rational number.
Given the equation tan(α) = √a, where 'a' is a rational number that is not a perfect square, we need to explore which of the trigonometric values (sin²α, tan²α, cos²α) is a rational number.
Key Relationships
- Trigonometric Identities:
- tan²α = sin²α / cos²α
- sin²α + cos²α = 1
Analysis of tan²α
- Calculate tan²α:
- tan²α = a (since tan(α) = √a)
- Since 'a' is rational, tan²α is rational.
Analysis of sin²α
- Using the identity:
- sin²α = tan²α * cos²α
- We know tan²α is rational, but cos²α must be evaluated to determine sin²α's rationality.
Analysis of cos²α
- Using the sine and cosine relationship:
- cos²α = 1 / (1 + tan²α)
- cos²α = 1 / (1 + a)
- Since 'a' is rational, 1 + a is also rational.
- Therefore, cos²α = 1 / (1 + a) is a rational number.
Conclusion
- Final Evaluation:
- sin²α depends on the value of cos²α, which is rational.
- tan²α is rational.
- cos²α is rational.
Thus, the correct answer is option 'C', as cos²α is the only confirmed rational number when tan(α) = √a and 'a' is a non-perfect square rational number.
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In tanα= √a, where a is a rational number which is not a perfect square, then which of the following is a rational number?a)sin2αb)tan2αc)cos2αd)noneCorrect answer is option 'C'. Can you explain this answer? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about In tanα= √a, where a is a rational number which is not a perfect square, then which of the following is a rational number?a)sin2αb)tan2αc)cos2αd)noneCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for In tanα= √a, where a is a rational number which is not a perfect square, then which of the following is a rational number?a)sin2αb)tan2αc)cos2αd)noneCorrect answer is option 'C'. Can you explain this answer?.
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