Final answer:
To solve the equation 3(sec^2θ + tan^2θ) = 5, we can use trigonometric identities to simplify the expression. The general solution for θ is 2nπ±(π/6).
Step-by-step explanation:
To solve the equation 3(sec^2θ + tan^2θ) = 5, we can simplify the expression using trigonometric identities. Let's start by using the identity sec^2θ = 1 + tan^2θ. Substituting this into the equation, we get 3(1 + tan^2θ + tan^2θ) = 5. Simplifying further, we have 3(2tan^2θ + 1) = 5. Dividing both sides by 3, we get 2tan^2θ + 1 = 5/3. Rearranging the equation, we get 2tan^2θ = 5/3 - 1. Combining the fractions, we have 2tan^2θ = (5 - 3)/3 = 2/3. Finally, dividing both sides by 2, we find that tan^2θ = 1/3.
Now, to find the general solution of θ, we need to find all values of θ that satisfy tan^2θ = 1/3. Taking the square root of both sides, we get tanθ = ±sqrt(1/3). Since tanθ is positive in the first and third quadrants, we can use the inverse tangent function to find the solutions: θ = tan^-1(sqrt(1/3)) and θ = 180° - tan^-1(sqrt(1/3)). These two solutions give us the general solution of θ.
Therefore, the correct answer is option B. 2nπ±(π/6).