Sure! Let's break down the process of identifying the phase shift of the trigonometric function given as:
[tex]\[
-2 \cos \left(\frac{1}{3} \theta - \frac{\pi}{3}\right) + 3
\][/tex]
Step-by-Step Solution:
1. Identify the general form of the cosine function:
A trigonometric function of the form [tex]\( \cos(bx - c) \)[/tex] has a phase shift determined by the fraction [tex]\(\frac{c}{b}\)[/tex], where [tex]\(b\)[/tex] and [tex]\(c\)[/tex] are constants in the given function.
2. Rewrite the function in the standard form:
In our function, [tex]\( \cos \left(\frac{1}{3} \theta - \frac{\pi}{3}\right) \)[/tex], compare this with [tex]\( \cos(bx - c) \)[/tex]:
- Here, [tex]\( b = \frac{1}{3} \)[/tex]
- And [tex]\( c = \frac{\pi}{3} \)[/tex]
3. Calculate the phase shift:
The phase shift formula is given by [tex]\(\frac{c}{b}\)[/tex]:
[tex]\[
\text{Phase shift} = \frac{c}{b} = \frac{\frac{\pi}{3}}{\frac{1}{3}} = \pi
\][/tex]
4. Determine the direction of the phase shift:
The direction of the phase shift is determined by the sign of [tex]\(c\)[/tex]:
- If [tex]\(c\)[/tex] is positive, the phase shift is to the right.
- If [tex]\(c\)[/tex] is negative, the phase shift is to the left.
In this case, [tex]\(c = \frac{\pi}{3}\)[/tex] which is positive.
Hence, the phase shift is [tex]\(\pi\)[/tex] units to the right.
Correct answer:
[tex]\[
\pi \text{ units to the right}
\][/tex]
