Final answer:
One solution to the trigonometric equation 3cosx - sin²x + 3 = 0 in the interval [0, 2π] is x = 0.
Step-by-step explanation:
Solution:
To find a solution to the trigonometric equation 3cosx - sin²x + 3 = 0 in the interval [0, 2π], we can solve the equation algebraically. Rearranging the equation, we have: sin²x = 3cosx - 3. Using the identity sin²x = 1 - cos²x, we can substitute to get: 1 - cos²x = 3cosx - 3. Rearranging, we have: cos²x + 3cosx - 4 = 0. Factoring, we get: (cosx + 4)(cosx - 1) = 0. Therefore, cosx + 4 = 0 or cosx - 1 = 0. Solving for x, we find two solutions: x = arccos(-4) and x = arccos(1).
In the interval [0, 2π], arccos(-4) is not defined, so the solution is x = arccos(1). This means x = 0, and therefore, one solution to the given trigonometric equation in the given interval is x = 0.