To find the exact value of [tex]\(\tan \theta\)[/tex] given the options:
A. [tex]\(\frac{\sqrt{5}}{3}\)[/tex]
B. [tex]\(\frac{3 \sqrt{14}}{14}\)[/tex]
we can compare their numerical values.
First, evaluate option A:
[tex]\[
\frac{\sqrt{5}}{3}
\][/tex]
Calculating the square root of 5:
[tex]\[
\sqrt{5} \approx 2.236
\][/tex]
Then divide by 3:
[tex]\[
\frac{2.236}{3} \approx 0.745
\][/tex]
So, the approximate numerical value of [tex]\(\frac{\sqrt{5}}{3}\)[/tex] is [tex]\(0.745\)[/tex].
Next, evaluate option B:
[tex]\[
\frac{3 \sqrt{14}}{14}
\][/tex]
Calculating the square root of 14:
[tex]\[
\sqrt{14} \approx 3.742
\][/tex]
Then multiply by 3:
[tex]\[
3 \cdot 3.742 \approx 11.226
\][/tex]
Finally, divide by 14:
[tex]\[
\frac{11.226}{14} \approx 0.802
\][/tex]
So, the approximate numerical value of [tex]\(\frac{3 \sqrt{14}}{14}\)[/tex] is [tex]\(0.802\)[/tex].
Given these results, the exact value of [tex]\(\tan \theta\)[/tex] is the one corresponding to the numerical value we found. Since [tex]\(\tan \theta\)[/tex] approximately equals [tex]\(0.802\)[/tex], the correct option is
B. [tex]\(\frac{3 \sqrt{14}}{14}\)[/tex]
