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If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the sides are proportional to
(a) 1:1:√2
(b) (1:√2:1
(c) √2:1:1
(d) 1:1:1?
(a) 1:1:√2
(b) (1:√2:1
(c) √2:1:1
(d) 1:1:1?
Most Upvoted Answer
If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the si...
Understanding the Given Condition
The equation given is:
cos A cos B + sin A sin B sin C = 1
This can be rewritten using the cosine of a sum formula. The relationship implies a specific configuration of angles in triangle ABC.
Analyzing the Angles
The expression can be interpreted as:
cos(A + B) = 1 - sin A sin B sin C
If cos(A + B) = 1, then A + B must equal 0. However, in a triangle, this suggests certain constraints on the angles.
Special Triangle Configuration
The maximum value of cos A and cos B occurs when angles A and B are 0 or 90 degrees. Therefore, the angles likely approach 45 degrees each to satisfy the equation.
Proportional Sides Analysis
Using the angles A = B = 45 degrees and C = 90 degrees, we can determine the side ratios:
- Opposite to angle A: 1 (adjacent)
- Opposite to angle B: 1 (adjacent)
- Opposite to angle C: √2 (hypotenuse)
Thus, the sides are in the ratio:
1:1:√2
Conclusion
Hence, the sides of triangle ABC are proportional to:
(a) 1:1:√2
This configuration satisfies the cosine condition provided in the problem statement.
The equation given is:
cos A cos B + sin A sin B sin C = 1
This can be rewritten using the cosine of a sum formula. The relationship implies a specific configuration of angles in triangle ABC.
Analyzing the Angles
The expression can be interpreted as:
cos(A + B) = 1 - sin A sin B sin C
If cos(A + B) = 1, then A + B must equal 0. However, in a triangle, this suggests certain constraints on the angles.
Special Triangle Configuration
The maximum value of cos A and cos B occurs when angles A and B are 0 or 90 degrees. Therefore, the angles likely approach 45 degrees each to satisfy the equation.
Proportional Sides Analysis
Using the angles A = B = 45 degrees and C = 90 degrees, we can determine the side ratios:
- Opposite to angle A: 1 (adjacent)
- Opposite to angle B: 1 (adjacent)
- Opposite to angle C: √2 (hypotenuse)
Thus, the sides are in the ratio:
1:1:√2
Conclusion
Hence, the sides of triangle ABC are proportional to:
(a) 1:1:√2
This configuration satisfies the cosine condition provided in the problem statement.
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If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the sides are proportional to(a) 1:1:√2(b) (1:√2:1(c) √2:1:1(d) 1:1:1?
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If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the sides are proportional to(a) 1:1:√2(b) (1:√2:1(c) √2:1:1(d) 1:1:1? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the sides are proportional to(a) 1:1:√2(b) (1:√2:1(c) √2:1:1(d) 1:1:1? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If in a triangle ABC, cos A cos B + sin A sin B sin C = 1, then the sides are proportional to(a) 1:1:√2(b) (1:√2:1(c) √2:1:1(d) 1:1:1?.
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