Question

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

Answer 1

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

\[
\cos(x) = \sin\left(x + \frac{\pi}{2}\right)
\]

Using this relationship, we first rewrite \( f(x) \):

\[
f(x) = \cos\left(\frac{1}{3}x - \frac{\pi}{6}\right)
\]

Now, we can express the cosine function in terms of the sine function:

\[
f(x) = \sin\left(\left(\frac{1}{3}x - \frac{\pi}{6}\right) + \frac{\pi}{2}\right)
\]

To combine the arguments, we have:

\[
f(x) = \sin\left(\frac{1}{3}x - \frac{\pi}{6} + \frac{\pi}{2}\right)
\]

Now, we need to find a common denominator for the terms inside the sine function. The angle \( \frac{\pi}{2} \) can be expressed as \( \frac{3\pi}{6} \):

\[
-\frac{\pi}{6} + \frac{3\pi}{6} = \frac{2\pi}{6} = \frac{\pi}{3}
\]

Therefore, we can rewrite our function as:

\[
f(x) = \sin\left(\frac{1}{3}x + \frac{\pi}{3}\right)
\]

Thus, the cosine function \( f(x) = \cos\left(\frac{1}{3}x - \frac{\pi}{6}\right) \) can be rewritten as the translated sine function:

\[
f(x) = \sin\left(\frac{1}{3}x + \frac{\pi}{3}\right)
\]