Let f ( ) = sin ( sin + sin 3 ) . Then f ( ) is [2000]

Question

Let $f (θ) = \sin θ (\sin θ + \sin 3 θ)$ . Then $f (θ)$ is [2000]

Smart Achievers · Accepted Answer

(3)

$f (θ) = \sin θ (\sin θ + \sin 3 θ)$

$= \sin θ (\sin θ + 3 \sin θ - 4 \sin^{3} θ)$

$= \sin θ (4 \sin θ - 4 \sin^{3} θ) = \sin^{2} θ (4 - 4 \sin^{2} θ)$

$= 4 \sin^{2} θ (1 - \sin^{2} θ)$

$= 4 \sin^{2} θ \cos^{2} θ = {(2 \sin θ \cos θ)}^{2} = {(\sin 2 θ)}^{2} \geq 0, which is true for all θ .$

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