FunctionHard
Question
Let f(θ) = sin θ (sin θ + sin 3 θ). The f (θ)
Options
A.≥ 0 only when θ ≥ 0
B.≤ 0 for all real θ
C.≥ 0 for all real θ
D.≤ 0 only when θ ≤ 0
Solution
It is given,
f(θ) = sin θ(sin θ + sin 3θ)
= (sin θ + 3sin θ - 4sin3θ) sin θ
= (4sin θ - 4sin 3θ) sin θ
= sin2 θ(4 - 4sin2 θ)
4sin2 θ cos2 θ = (2sin2 θ cos2 θ)
= (sin 2θ)2 ≥ 0
which is true for all θ
f(θ) = sin θ(sin θ + sin 3θ)
= (sin θ + 3sin θ - 4sin3θ) sin θ
= (4sin θ - 4sin 3θ) sin θ
= sin2 θ(4 - 4sin2 θ)
4sin2 θ cos2 θ = (2sin2 θ cos2 θ)
= (sin 2θ)2 ≥ 0
which is true for all θ
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