Trigonometry problem

Find the measures of angles x at which the function f(x) = -(2/3)cos(x - π/2) + 3 is at its maximum value.

A. -π/2 ± nπ, n an integer

B. 3π/2 ± 2nπ, n any real number

C. -π/2 ± 2nπ, n an integer

D. 3π/2 ± nπ, n any real number

The maximum value for this function will be 3 + 2/3 = 11/3. We know this from the fact that the minimum for cos(x - π/2) is -1, and -1(-2/3) = 2/3. 

If cos(x - π/2) = -1, then x - π/2 = π, so that x = π + π/2 = 3π/2. Now, since cos(x) shows the same value with angle increments of 2π, cos(x - π/2) function will also show the same certain value with angle increments of 2π. This means that our x can be 3π/2, 3π/2 - 2π, 3π/2 + 2π, 3π/2 - 4π, 3π/2 + 4π, etc. This means that angles x produce the same function values for f(x) = -(2/3)cos(x - π/2) + 3 at 3π/2 + 2nπ, 3π/2 - 2nπ, where n is an integer. Thus the angles with the maximum values are 3π/2 ± 2nπ, n an integer, and this is the same thing as the answer (C) shows, since 3π/2 - (-π/2) = 2π. Choice (B) is not correct because n must be an integer, otherwise the angles will not produce the same value, and choices (A) and (D) also fail for these reasons as well as adding and subtracting the wrong angle measures. You may plug in each answer choice to make an educated guess, but you may get confused with angle additions if you are not familiar with this topic.

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