To find the value of [tex]\(\tan \theta\)[/tex] given that [tex]\(\cos \theta = \frac{5}{13}\)[/tex], let's follow these steps:
1. Identify the sides of the right triangle:
- Recall that in the context of a right triangle, [tex]\(\cos \theta\)[/tex] is the ratio of the length of the adjacent side to the hypotenuse. Here, [tex]\(\cos \theta = \frac{5}{13}\)[/tex] means:
- Adjacent side = 5
- Hypotenuse = 13
2. Use the Pythagorean theorem to find the opposite side:
- The Pythagorean theorem states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, we set up the equation:
[tex]\[
\text{Hypotenuse}^2 = \text{Adjacent}^2 + \text{Opposite}^2
\][/tex]
- Plug in the known values:
[tex]\[
13^2 = 5^2 + \text{Opposite}^2
\][/tex]
- Simplify the equation:
[tex]\[
169 = 25 + \text{Opposite}^2
\][/tex]
- Solve for the opposite side:
[tex]\[
\text{Opposite}^2 = 169 - 25
\][/tex]
[tex]\[
\text{Opposite}^2 = 144
\][/tex]
[tex]\[
\text{Opposite} = \sqrt{144} = 12
\][/tex]
3. Calculate [tex]\(\tan \theta\)[/tex]:
- Recall that [tex]\(\tan \theta\)[/tex] is the ratio of the opposite side to the adjacent side. With the values we have:
- Opposite side = 12
- Adjacent side = 5
- So,
[tex]\[
\tan \theta = \frac{12}{5} = 2.4
\][/tex]
Therefore, [tex]\(\tan \theta = 2.4\)[/tex].
