To find [tex]\(\tan \theta\)[/tex] given that [tex]\(\sin \theta = \frac{9}{15}\)[/tex], follow these steps:
1. Simplify [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \theta = \frac{9}{15} = \frac{3}{5}
\][/tex]
2. Use the Pythagorean identity to find [tex]\(\cos \theta\)[/tex]:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
Substitute [tex]\(\sin \theta\)[/tex]:
[tex]\[
\left(\frac{3}{5}\right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\frac{9}{25} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{25}{25} - \frac{9}{25}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{16}{25}
\][/tex]
[tex]\[
\cos \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}
\][/tex]
3. Calculate [tex]\(\tan \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
Substitute the values of [tex]\(\sin \theta\)[/tex] and [tex]\(\cos \theta\)[/tex]:
[tex]\[
\tan \theta = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{5} \div \frac{4}{5} = \frac{3}{5} \times \frac{5}{4} = \frac{3}{4}
\][/tex]
The value of [tex]\(\tan \theta\)[/tex] is [tex]\(\frac{3}{4}\)[/tex].
Now, simplify [tex]\(\frac{3}{4}\)[/tex] to match the given choices:
[tex]\[
\frac{3}{4} = \frac{9}{12}
\][/tex]
So, [tex]\( \tan \theta = \frac{9}{12} \)[/tex].
Thus, the correct answer is:
[tex]\[
\boxed{\frac{9}{12}}
\][/tex]
