To solve for [tex]\(\tan \theta\)[/tex] given that [tex]\(\cos \theta = \frac{3}{5}\)[/tex], follow these steps:
1. Recall the Pythagorean identity for sine and cosine:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
2. Substitute [tex]\(\cos \theta = \frac{3}{5}\)[/tex] into the identity:
[tex]\[
\sin^2 \theta + \left( \frac{3}{5} \right)^2 = 1
\][/tex]
3. Square the cosine value:
[tex]\[
\sin^2 \theta + \frac{9}{25} = 1
\][/tex]
4. Subtract [tex]\(\frac{9}{25}\)[/tex] from 1:
[tex]\[
\sin^2 \theta = 1 - \frac{9}{25}
\][/tex]
5. Convert 1 to a fraction with a denominator of 25:
[tex]\[
\sin^2 \theta = \frac{25}{25} - \frac{9}{25} = \frac{16}{25}
\][/tex]
6. Take the square root of both sides to find [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \theta = \sqrt{\frac{16}{25}} = \frac{4}{5}
\][/tex]
Since we are not given specific details about the quadrant, we assume [tex]\(\sin \theta\)[/tex] is positive.
7. Recall the definition of tangent:
[tex]\[
\tan \theta = \frac{\sin \theta}{\cos \theta}
\][/tex]
8. Substitute the known values of sine and cosine:
[tex]\[
\tan \theta = \frac{\frac{4}{5}}{\frac{3}{5}}
\][/tex]
9. Simplify the fraction:
[tex]\[
\tan \theta = \frac{4}{5} \times \frac{5}{3} = \frac{4}{3}
\][/tex]
Therefore, [tex]\(\tan \theta = \frac{4}{3}\)[/tex].
