Answer:
To solve the equation 8cos(x) + 16cos(x) + 8 = 0 over the interval [0, 2x), we can combine the cosine terms:
8cos(x) + 16cos(x) + 8 = 0
24cos(x) + 8 = 0
24cos(x) = -8
cos(x) = -8/24
cos(x) = -1/3
Now, to find the solutions over the interval [0, 2x), we need to consider the values of x that satisfy cos(x) = -1/3.
Using the inverse cosine function, we can find the principal solution:
x = arccos(-1/3)
The principal solution gives us one solution within the interval [0, π]. However, since we are looking for solutions within the interval [0, 2x), we need to consider other angles that satisfy the equation within this interval.
To do that, we can use the periodicity of the cosine function. We know that the cosine function repeats itself every 2π. So, if x = arccos(-1/3) is a solution within [0, π], then x + 2πn (where n is an integer) will also be a solution within [0, 2x).
Therefore, the exact solutions over the interval [0, 2x) are:
x = arccos(-1/3) + 2πn, where n is an integer.
Please note that the specific values of x depend on the exact value of arccos(-1/3) and the integer values of n.
Explanation: