2. Solve: [tex]$\left(0^{\circ} \leq \theta \leq 180^{\circ}\right)$[/tex]
a. [tex]$4 \sin \theta=3 \operatorname{cosec} \theta$[/tex]
Answer (1)
Rewrite cosec θ as s i n θ 1 , leading to 4 sin θ = s i n θ 3 .
Multiply both sides by sin θ to get 4 sin 2 θ = 3 , then divide by 4 to obtain sin 2 θ = 4 3 .
Take the square root to find sin θ = ± 2 3 , but since 0 ∘ ≤ θ ≤ 18 0 ∘ , consider only the positive root sin θ = 2 3 .
Identify the angles θ in the given range that satisfy sin θ = 2 3 , which are 6 0 ∘ , 12 0 ∘ .
Explanation
Problem Analysis We are given the equation 4 sin θ = 3 cosec θ and asked to solve for θ in the interval 0 ∘ ≤ θ ≤ 18 0 ∘ .
Rewrite the equation First, we rewrite the equation using the definition of cosecant: cosec θ = s i n θ 1 . The equation becomes 4 sin θ = sin θ 3 .
Multiply by sin θ Next, we multiply both sides of the equation by sin θ to get 4 sin 2 θ = 3.
Divide by 4 Then, we divide both sides by 4 to obtain sin 2 θ = 4 3 .
Take the square root Taking the square root of both sides gives sin θ = ± 2 3 .
Consider the range of θ Since 0 ∘ ≤ θ ≤ 18 0 ∘ , sin θ must be non-negative. Therefore, we only consider the positive root: sin θ = 2 3 .
Find the angles We need to find the angles θ in the interval [ 0 ∘ , 18 0 ∘ ] such that sin θ = 2 3 . We know that sin 6 0 ∘ = 2 3 . Also, since sin ( 18 0 ∘ − x ) = sin x , we have sin ( 18 0 ∘ − 6 0 ∘ ) = sin 12 0 ∘ = 2 3 . Thus, the solutions are θ = 6 0 ∘ and θ = 12 0 ∘ .
Final Answer Therefore, the solutions to the equation 4 sin θ = 3 cosec θ in the interval 0 ∘ ≤ θ ≤ 18 0 ∘ are θ = 6 0 ∘ and θ = 12 0 ∘ .
Examples
Imagine you're designing a solar panel that needs to capture the maximum amount of sunlight. The angle at which the sun's rays hit the panel affects its efficiency. By solving trigonometric equations like the one above, you can determine the optimal angles for the panel to maximize energy capture throughout the day. This ensures that the solar panel operates at its peak performance, converting sunlight into electricity as efficiently as possible. Understanding trigonometric relationships is crucial in optimizing renewable energy systems.
