If A 0 ,0 , B θ , cos ⁡ θ and C sin 3 ⁡ θ , 0 are the vertices of a triangle A B C , then the value of θ for which the triangle has the maximum area is where θ ∈ 0 , π 2

Step-by-step solution

Area of triangle A B C , △ = 1 2 0 0 1 θ cos ⁡ θ 1 sin 3 ⁡ θ 0 1 ⇒ △ = 1 2 sin 3 ⁡ θ cos ⁡ θ d △ d θ = 3 sin 2 ⁡ θ cos 2 ⁡ θ - sin 4 ⁡ θ 2 = 0 ⇒ sin 2 ⁡ θ 3 cos 2 ⁡ θ - sin 2 ⁡ θ = 0 ⇒ sin ⁡ θ = 0 or tan 2 ⁡ θ = 3 θ = 0 or θ = π 3 d 2 △ d θ 2 < 0   f o r   θ = π 3