The graph of the function y = sin4x is identical to the function y = cos4(x − p). Find the smallest possible positive value of p, as a fraction in terms of pi. Explain your thinking.

Answer 1

Find the p such that sin(4x) = cos(4x - 4p) = cos4x cos4p + sin4x sin 4p

Clearly, if sin 4p = 1 and cos 4p = 0, we have equality . As, when sin 4p = 1, cos 4p = 0, we must only consider the first equation

sin 4p = 1 at 4p = pi/2, pi/2 + 2 pi, ... Clearly, the smallest value is 4p = pi/2, or p = pi/8

Now, we could have reasoned this out in another way. The cos curve is pi/2 ahead of the sin curve, as cos 0 = sin pi/2, cos pi/2 = sin pi, ... Thus, sin x = cos(x - pi/2); then sin 4x = cos(4x - pi/2) = cos(4(x - pi/8))

Answer 2

sin 4x = cos (4x - 4p)

sin 4x = sin(π/2 - (4x - 4p))

sin 4x = sin(π/2 - 4x + 4p)

4x = π/2 - 4x + 4p

8x = π/2 + 4p

8x - π/2 = 4p

2x - π/8 = p