Final answer:
To solve the equation 4sinθ + 3cosθ = 0, use the trigonometric identity sinθ = sqrt(1 - cos^2θ), substitute it into the equation, and simplify to find the value of cos^2θ.
Step-by-step explanation:
To solve the equation 4sinθ + 3cosθ = 0, we can use the trigonometric identity sinθ = sqrt(1 - cos^2θ). Substituting this into the equation, we get 4(sqrt(1 - cos^2θ)) + 3cosθ = 0. Simplifying further, we have 4sqrt(1 - cos^2θ) = -3cosθ. Squaring both sides of the equation, we get 16(1 - cos^2θ) = 9cos^2θ. Rearranging and combining like terms, we have 25cos^2θ + 16cos^2θ = 16. Solving for cos^2θ, we get cos^2θ = 16/41.