Question

rewrite the cosine function f (x) = cos (1/3 x - pi/6) as a translated sine function

Answer 1

Use cos θ = sin(θ + π/2). With θ = (1/3)x − π/6:

f(x) = cos((1/3)x − π/6) = sin((1/3)x − π/6 + π/2)
= sin((1/3)x + π/3).

Equivalently, f(x) = sin((1/3)(x + π)), a sine function translated left by π.