Final answer:
When solving the equation 16cos(15x) + 8 = 2, we isolate cos(15x) to find it equals -3/8. The solutions to the equation in terms of 'x' are x = ± 0.130n * (2π/15), so the expressions that represent all solutions are e. 0.130n * 2π/15 and f. 1.955n * 2π.
Step-by-step explanation:
The equation we need to solve is 16cos(15x) + 8 = 2. The first step is to isolate the cosine term:
16cos(15x) = 2 - 8
16cos(15x) = -6
cos(15x) = -6 / 16
cos(15x) = -3/8
Now, we look for solutions where the cosine function yields -3/8. Since the cosine function is periodic, there will be an infinite number of solutions, and they can be represented using the general form x = (1/n) * 2πk + α, where k is an integer, α is the principal value of the angle (our specific solution), and π is pi.
Using the inverse cosine function on a calculator, we find that the principal value that gives cos(x) = -3/8 is approximately 1.955 or 0.130 radians (in the positive and negative direction, since cosine is even). Therefore, we have:
x = ± 1.955 + 2πk/15, since 15x is the argument in the original equation
Adjusting for the argument of cosine, we divide the principal values by 15 to obtain the solutions expressed in 'x':
x = ± 0.130n * (2π/15)
So the options representing all solutions to the equation are e. 0.130n⋅ 2π/15 and f. 1.955n⋅ 2π.