Final answer:
To solve 2cos(5x) + 1 = 0, we find the values of 5x where cos(5x) is -1/2. The correct answer, given the options, is C) 5π/3, which simplifies to x = π/3.
Step-by-step explanation:
The question requires solving the trigonometric equation 2cos(5x) + 1 = 0. To find the solution, we need to find the possible values of 5x for which cosine equals -1/2, because cos(5x) will equal -1/2 when 2cos(5x) + 1 equals 0.
This equation is satisfied if the argument of the cosine is an integral multiple of π/2, 3π/2, 5π/2, and so on. However, since cosine of -1/2 is related to the angles 120° (or 2π/3) and 240° (or 4π/3), we look for solutions where 5x equals 2π/3 or 5x equals 4π/3.
For the first case, when 5x equals 2π/3, we can solve for x to get x = (2π/3)/5. This simplifies to x = π/3. For the second case, when 5x equals 4π/3, we solve for x to get x = (4π/3)/5.
Correct Answer
Checking against the choices provided, the correct option is C) 5π/3, as it represents one of the solutions where 5x equals 4π/3, and simplifies to x = π/3.