Question

Let f(∅) = sino + 2sin2∅- sin3∅ - 3 then in 0 € (0, π) number of solutions of f(∅) = 0 is
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Answer 1

Answer:

f(θ)=sinθ(sinθ+sin3θ)

⇒f(θ)=sinθ(2sin(2θ+3θ)cos(2θ−3θ))[∵Applying identity:sinA+sinB=2sin2A+Bcos2A−B]

⇒f(θ)=sinθ(2sin2θcosθ)[∵cos(−θ)=cosθ]

⇒f(θ)=2sinθcosθsin2θ

⇒f(θ)=sin22θ[∵Applying identity:sin2θ=2sinθcosθ]

∴f(θ)≥0 for all real θ and lies in region [0,1].

Hence, f(θ)≥0 for all real θ.