Question

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

A. f(x)=sin(1/3x+pi/3)
B. f(x) = sin(1/3(x+pi))
C. f(x) = sin (1/3x+pi)
D. f(x) = sin (1/3x)

Answer 1

To rewrite the function \( f(x) = \cos\left(\frac{1}{3}x - \frac{\pi}{6}\right) \) as a translated cosine function, we can use the phase shift identity between cosine and sine.

Recall that:
\[
\cos\left(x - \frac{\pi}{2}\right) = \sin(x)
\]

We can express \( \cos\left(\frac{1}{3}x - \frac{\pi}{6}\right) \) in terms of sine:
\[
\cos\left(\frac{1}{3}x - \frac{\pi}{6}\right) = \sin\left(\frac{1}{3}x - \frac{\pi}{6} + \frac{\pi}{2}\right)
\]

Now, we need to simplify \(-\frac{\pi}{6} + \frac{\pi}{2}\):
\[
-\frac{\pi}{6} + \frac{\pi}{2} = -\frac{\pi}{6} + \frac{3\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}
\]

Thus, we can rewrite \( f(x) \) as:
\[
f(x) = \sin\left(\frac{1}{3}x + \frac{\pi}{3}\right)
\]

The correct option from the choices given is:
**A. f(x) = sin(1/3x + pi/3)**.