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The least value of 2 sin2 θ + 3 cos2 θ is (SSC CGL 1st Sit. 2012)
- a)3
- b)5
- c)1
- d)2
Correct answer is option 'D'. Can you explain this answer?
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The least value of 2 sin2 θ + 3 cos2 θ is (SSC CGL 1st Si...
2sin2 θ + 3 cos2 θ = 2 sin2 θ + 2cos2 θ + cos2 θ
= 2 (sin2θ + cos2θ) + cos2θ = 2 + cos2θ
∴ Least value = 2 + 0 = 2 [∵ cos2 θ ≥ 0]
= 2 (sin2θ + cos2θ) + cos2θ = 2 + cos2θ
∴ Least value = 2 + 0 = 2 [∵ cos2 θ ≥ 0]
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The least value of 2 sin2 θ + 3 cos2 θ is (SSC CGL 1st Si...
Given Expression: 2 sin^2θ + 3 cos^2θ
Strategy: We need to minimize the given expression by using trigonometric identities.
- Trigonometric Identity: sin^2θ + cos^2θ = 1
- Conversion: We can convert the given expression using the above identity.
- Conversion: 2 sin^2θ + 3 cos^2θ = 2(1 - cos^2θ) + 3 cos^2θ
- Simplify: 2 - 2cos^2θ + 3cos^2θ
- Simplify: 2 + cos^2θ
- Minimum Value: The minimum value of cos^2θ is 0.
- Minimum Value: Therefore, the least value of the expression is 2 + 0 = 2.
- Final Answer: The least value of 2 sin^2θ + 3 cos^2θ is 2.
Therefore, option 'D' is the correct answer.
Strategy: We need to minimize the given expression by using trigonometric identities.
- Trigonometric Identity: sin^2θ + cos^2θ = 1
- Conversion: We can convert the given expression using the above identity.
- Conversion: 2 sin^2θ + 3 cos^2θ = 2(1 - cos^2θ) + 3 cos^2θ
- Simplify: 2 - 2cos^2θ + 3cos^2θ
- Simplify: 2 + cos^2θ
- Minimum Value: The minimum value of cos^2θ is 0.
- Minimum Value: Therefore, the least value of the expression is 2 + 0 = 2.
- Final Answer: The least value of 2 sin^2θ + 3 cos^2θ is 2.
Therefore, option 'D' is the correct answer.
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The least value of 2 sin2 θ + 3 cos2 θ is (SSC CGL 1st Sit. 2012)a)3b)5c)1d)2Correct answer is option 'D'. Can you explain this answer?
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