To solve the problem, we need to verify which of the provided options correctly yields the value of [tex]\(\tan \theta\)[/tex]. We will consider both options step-by-step and evaluate them.
1. Option A: [tex]\(\frac{\sqrt{5}}{3}\)[/tex]
- First, calculate [tex]\(\sqrt{5}\)[/tex]:
[tex]\[
\sqrt{5} \approx 2.236
\][/tex]
- Next, divide this result by 3:
[tex]\[
\frac{\sqrt{5}}{3} \approx \frac{2.236}{3} \approx 0.745
\][/tex]
2. Option B: [tex]\(3 \sqrt{14}\)[/tex]
- First, calculate [tex]\(\sqrt{14}\)[/tex]:
[tex]\[
\sqrt{14} \approx 3.742
\][/tex]
- Next, multiply this result by 3:
[tex]\[
3 \sqrt{14} \approx 3 \times 3.742 = 11.224
\][/tex]
Comparing both values obtained from options A and B:
- Option A yields approximately [tex]\(0.745\)[/tex].
- Option B yields approximately [tex]\(11.224\)[/tex].
Given these results, the correct value of [tex]\(\tan \theta\)[/tex] corresponds to one of the approximated values. Since [tex]\(0.745\)[/tex] and [tex]\(11.224\)[/tex] are the outcomes, we conclude that:
- [tex]\(\frac{\sqrt{5}}{3} \approx 0.745\)[/tex]
- [tex]\(3 \sqrt{14} \approx 11.224\)[/tex]
Therefore, depending on which numerical value is required, [tex]\(\tan \theta\)[/tex] can be associated with:
- [tex]\(\frac{\sqrt{5}}{3}\)[/tex] if [tex]\(\tan \theta \approx 0.745\)[/tex]
- [tex]\(3 \sqrt{14}\)[/tex] if [tex]\(\tan \theta \approx 11.224\)[/tex]
