Given [tex]\(\cos \theta = \frac{3}{5}\)[/tex], we are to find [tex]\(\tan \theta\)[/tex].
1. Let's consider a right triangle where [tex]\(\cos \theta = \frac{3}{5}\)[/tex]. Recall that [tex]\(\cos \theta\)[/tex] is defined as the ratio of the adjacent side to the hypotenuse. So, we have:
- Adjacent side = 3
- Hypotenuse = 5
2. To find the opposite side, we use the Pythagorean theorem which states:
[tex]\[
\text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2
\][/tex]
Substituting the known values:
[tex]\[
5^2 = 3^2 + \text{Opposite}^2
\][/tex]
Simplifying this equation, we get:
[tex]\[
25 = 9 + \text{Opposite}^2
\][/tex]
Solving for the opposite side:
[tex]\[
\text{Opposite}^2 = 25 - 9 = 16
\][/tex]
Taking the square root of both sides, we find:
[tex]\[
\text{Opposite} = 4
\][/tex]
3. Now, we use the definition of [tex]\(\tan \theta\)[/tex], which is the ratio of the opposite side to the adjacent side:
[tex]\[
\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{4}{3}
\][/tex]
Therefore, [tex]\(\tan \theta = \frac{4}{3}\)[/tex].
