Final answer:
To solve the equation 3cos^2(x) + 10sin^2(x) = 0, substitute the trigonometric identity cos^2(x) + sin^2(x) = 1, simplify the equation, solve the quadratic equation for sin^2(x), and find the values of x using the inverse sine function.
Step-by-step explanation:
To solve the equation 3cos2(x) + 10sin2(x) = 0, we can use the trigonometric identity cos2(x) + sin2(x) = 1. We can substitute this identity into the equation to get 3(1 - sin2(x)) + 10sin2(x) = 0. Simplifying, we have 3 - 3sin2(x) + 10sin2(x) = 0. Combining like terms, we get 7sin2(x) - 3 = 0.
Next, we can solve this quadratic equation for sin2(x). Dividing both sides of the equation by 7, we have sin2(x) = 3/7. Taking the square root of both sides, we get sin(x) = ±√(3/7).
Finally, we can find the values of x by using the inverse sine function. The solutions for 0 ≤ x ≤ 360 are x = arcsin(√(3/7)) ≈ 38.21° and x = 180 - arcsin(√(3/7)) ≈ 141.79°. Therefore, the correct answer is B. x = 180.