Final answer:
To satisfy the equation 4 cos(x) - 2 = 0 within the interval [0, 2], we find that x = π/3 (approximately 1.0472) is the only solution.
Step-by-step explanation:
The equation given is 4 cos(x) - 2 = 0. To find the values of x that satisfy this equation within the interval [0, 2], we first solve for cos(x):
4 cos(x) = 2
cos(x) = 2/4
cos(x) = 1/2
Now, we need to find the values of x for which the cosine is 1/2. The cosine function has the value of 1/2 at x = π/3 and x = 5π/3 within the range of 0 to 2π. However, since we are only interested in the interval [0, 2], we need the equivalent values in this interval.
The value of π/3 is approximately 1.0472, which is within the interval [0, 2]. The value of 5π/3 is approximately 5.23599, which is outside our interval of interest. Therefore, the only value in the given interval that satisfies the equation is x = π/3.
So, the solution to the equation within the interval [0, 2] is x = π/3 (rounded to four decimal places, x ≈ 1.0472).