Final answer:
The equation 2cosx + 1 = 0 can be solved by isolating cosx to get cosx = -1/2. The general solution is x = 2nπ ± 2π/3, where n is an integer, representing the angles in radians where the cosine value is -1/2.
Step-by-step explanation:
To solve the equation 2cosx + 1 = 0, we first isolate the cosine term so that we can find the general solution for x. We rewrite the equation as cosx = -1/2. The general solution for when cosine equals -1/2 is given by x = 2nπ ± 2π/3, where n is an integer. This represents the angles in radians for which the cosine value is -1/2 at different points in the sine wave cycle. For example, the cosine of 2π/3 and 4π/3 equals -1/2 in the unit circle, corresponding to the angles in the second and third quadrants, respectively. Since the cosine function is periodic with period 2π, the general solution includes adding 2nπ to account for all the rotations around the circle where the cosine of the angle will still be -1/2.