Final answer:
To change the period of the cosine function from 2π to π/2, multiply the variable x by 4, resulting in the function f(x) = cos(4x). The general formula for period T in a cosine function is T = 2π/B; hence for a period of π/2, B equals 4.
Step-by-step explanation:
Modifying the Period of a Cosine Function
The cosine function f(x) = cos(x) has a period of 2π. To modify the period to become π/2, we need to alter the function to affect its rate of oscillation. This is achieved by changing the frequency of the function, which is inversely related to the period. Specifically, since the desired period (π/2) is 1/4th of the original period (2π), we would need to increase the frequency of the function by a factor of 4. The modified function with the desired period is f(x) = cos(4x).
To understand why this modification works, consider the general form of the cosine function f(x) = A cos(Bx + C) + D, where A is the amplitude, B affects the period of the function, C represents the phase shift, and D is the vertical shift. The period T of the cosine function is given by T = 2π/B. Therefore, to find B for a period of π/2, we rearrange the formula to B = 2π/T and substitute T with π/2, yielding B = 4. Hence, the modified function becomes f(x) = cos(4x), as previously stated.
Examples in physics, such as a block on a spring undergoing simple harmonic motion (SHM), can be modeled with a cosine function. The function x(t) = A cos(omega t + phi) describes the position x of the block over time t, where A is the amplitude, omega is the angular frequency, and phi is the phase shift. If the block's motion had to repeat four times more frequently, the omega value would increase by a factor of four, similarly to how we adjusted B in the cosine function to change its period.