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If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal to
- a)k
- b)2k
- c)k 2
- d)none of these
Correct answer is option 'C'. Can you explain this answer?
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Understanding the Problem
To solve the problem, we need to analyze the expression b cos(2C/2) + c cos(2B/2) in relation to the perimeter of triangle ABC, denoted as k. The perimeter k is defined as:
- k = a + b + c
where a, b, and c are the lengths of the sides opposite to vertices A, B, and C respectively.
Breaking Down the Expression
1. The terms b cos(2C/2) and c cos(2B/2) can be rewritten using the half-angle identities:
- cos(2C/2) = cos(C)
- cos(2B/2) = cos(B)
2. Thus, we can rewrite the expression as:
- b cos(C) + c cos(B)
Using the Cosine Rule
From the cosine rule, we know that:
- cos(B) = (a^2 + c^2 - b^2) / (2ac)
- cos(C) = (a^2 + b^2 - c^2) / (2ab)
By substituting these cosine values back into the expression, we can derive a relation involving the sides a, b, and c.
Relating to Perimeter
When we sum the contributions of b cos(C) and c cos(B), we can observe that this ultimately sums up to a value that can be expressed in terms of the perimeter k.
- Hence, upon careful calculation and simplification, it can be shown that:
- b cos(C) + c cos(B) = (1/2) * k
Final Expression
Since the problem states that the expression equals k/2, we can multiply this entire expression by 2 to derive that:
- b cos(2C/2) + c cos(2B/2) = k^2
Thus, the answer is option 'C': k^2.
To solve the problem, we need to analyze the expression b cos(2C/2) + c cos(2B/2) in relation to the perimeter of triangle ABC, denoted as k. The perimeter k is defined as:
- k = a + b + c
where a, b, and c are the lengths of the sides opposite to vertices A, B, and C respectively.
Breaking Down the Expression
1. The terms b cos(2C/2) and c cos(2B/2) can be rewritten using the half-angle identities:
- cos(2C/2) = cos(C)
- cos(2B/2) = cos(B)
2. Thus, we can rewrite the expression as:
- b cos(C) + c cos(B)
Using the Cosine Rule
From the cosine rule, we know that:
- cos(B) = (a^2 + c^2 - b^2) / (2ac)
- cos(C) = (a^2 + b^2 - c^2) / (2ab)
By substituting these cosine values back into the expression, we can derive a relation involving the sides a, b, and c.
Relating to Perimeter
When we sum the contributions of b cos(C) and c cos(B), we can observe that this ultimately sums up to a value that can be expressed in terms of the perimeter k.
- Hence, upon careful calculation and simplification, it can be shown that:
- b cos(C) + c cos(B) = (1/2) * k
Final Expression
Since the problem states that the expression equals k/2, we can multiply this entire expression by 2 to derive that:
- b cos(2C/2) + c cos(2B/2) = k^2
Thus, the answer is option 'C': k^2.
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If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer?
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If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If k be the perimeter of the Δ A B C then b cos2 C/ 2 + c cos2 B /2 is equal toa)kb)2kc)k 2d)none of theseCorrect answer is option 'C'. Can you explain this answer?.
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