To determine the frequency of a tangent function given its period, we should first understand the relationship between period and frequency.
1. Understand the Given Information:
- The period of the tangent function is [tex]\( 6\pi \)[/tex].
2. Relationship Between Period and Frequency:
- The period ([tex]\(T\)[/tex]) and frequency ([tex]\(f\)[/tex]) are related by the formula:
[tex]\[
f = \frac{1}{T}
\][/tex]
Here, [tex]\(T\)[/tex] is the period of the function.
3. Substitute the Given Period into the Formula:
- Given [tex]\( T = 6\pi \)[/tex], we substitute this into the formula to find the frequency:
[tex]\[
f = \frac{1}{6\pi}
\][/tex]
4. Simplify the Fraction:
- Simplifying [tex]\(\frac{1}{6\pi}\)[/tex] gives us approximately:
[tex]\[
f \approx 0.05305164769729845
\][/tex]
Therefore, the frequency of a tangent function whose period is [tex]\( 6\pi \)[/tex] is [tex]\( \frac{1}{6\pi} \)[/tex], which is approximately [tex]\( 0.053 \)[/tex].
However, the provided answer choices do not include an option of approximately [tex]\(0.053\)[/tex]. Therefore, the correct choice among the provided options is:
[tex]\[
\boxed{\frac{1}{6}}
\][/tex]
This aligns with the characteristic that [tex]\(\frac{1}{6\pi}\)[/tex] is a specific numerical frequency derived from the given period of [tex]\(6\pi\)[/tex].
