JEE Main 2023Trigonometric EquationsTrigonometric EquationsMediumQuestionLet ∣cosθcos(60−θ)cos(60+θ)∣≤18,θϵ[0,2π]|\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8}, \theta \epsilon[0,2 \pi]∣cosθcos(60−θ)cos(60+θ)∣≤81,θϵ[0,2π]. Then, the sum of all θ∈[0,2π]\theta \in[0,2 \pi]θ∈[0,2π], where cos3θ\cos 3 \thetacos3θ attains its maximum value, is :OptionsA6π6 \pi6πB9π9 \pi9πC18π18 \pi18πD15π15 \pi15πCheck AnswerHide SolutionSolution∣cosθcos(60−θ)cos(60+θ)∣≤18⇒14∣cos3θ∣≤18cos3θ is max if cos3θ=12∴θ=π9,5π9,7π9,11π9,13π9,17π9∑θi=6π\begin{aligned} & |\cos \theta \cos (60-\theta) \cos (60+\theta)| \leq \frac{1}{8} \\ & \Rightarrow \frac{1}{4}|\cos 3 \theta| \leq \frac{1}{8} \\ & \cos 3 \theta \text { is max if } \cos 3 \theta=\frac{1}{2} \\ & \therefore \theta=\frac{\pi}{9}, \frac{5 \pi}{9}, \frac{7 \pi}{9}, \frac{11 \pi}{9}, \frac{13 \pi}{9}, \frac{17 \pi}{9} \\ & \sum \theta_i=6 \pi \end{aligned}∣cosθcos(60−θ)cos(60+θ)∣≤81⇒41∣cos3θ∣≤81cos3θ is max if cos3θ=21∴θ=9π,95π,97π,911π,913π,917π∑θi=6π