Trigonometric EquationMedium
Question
Minimum value of the expression cos2 θ -(6 sin θ cos θ) + 3 sin2 θ + 2, is -
Options
A.4 + √10
B.4 - √10
C.0
D.4
Solution
Minimum value of the expression
cos2θ - (6 sinθ cosθ) + 3sin2θ + 2
= 1 + 2sin2θ - 6sinθ cosθ + 2
= 3 + 1 - cos2θ - 3sin2θ
= 4 - (cos2θ + 3 sin 2θ)
∵ -
≤ cos2θ + 3 sin 2θ ≤ 
≥ - (cos2θ + 3sin 2θ) ≥ - 
4 +
≥ 4 - (cos2θ + 3sin 2θ) ≥ 4 - 
4 -
≤ 4 - (cos2θ + 3sin 2θ) ≤ 4 + 
minimum value is 4 -
cos2θ - (6 sinθ cosθ) + 3sin2θ + 2
= 1 + 2sin2θ - 6sinθ cosθ + 2
= 3 + 1 - cos2θ - 3sin2θ
= 4 - (cos2θ + 3 sin 2θ)
∵ -
≤ cos2θ + 3 sin 2θ ≤ ≥ - (cos2θ + 3sin 2θ) ≥ - 4 +
≥ 4 - (cos2θ + 3sin 2θ) ≥ 4 - 4 -
≤ 4 - (cos2θ + 3sin 2θ) ≤ 4 + minimum value is 4 -
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