Identify the horizontal middle of a sample period for the function: (Please help no one answers my questions)

Question

f (t) = 4sec(3t − 6π)

 

a. x=4pi/3

b. x=5pi/2

c. x=pi

d. x=3+pi

Answer 1

We can use the general equation for a secant function, f(x) = α sec (βx - c) - d, to help us solve this problem. In this case, α=4, β=3, c=6π, d=0. The formula for finding the period of secant functions (as well as sin, cos, and cosec) is period=2π / |β|. Therefore the period is 2π/3. It is not entirely clear what exactly you mean by "horizontal middle of a sample period," but if we assume the sample period starts at x=0, then it will extend to x=2π/3 with a middle point of x=π/3. The next sample period will then start at x=2π/3 and extend to 4π/3 with a middle of x=3π/3 or π. Looking back to our possible answers we see (c) is the right answer. If we wanted to extend this farther, we would see that the endpoints of our sample periods will be any even number times π/3, and the middles of our sample periods will be any odd number times π/3.