Sure, let's work through the problem step-by-step.
We are given a point [tex]\((3, -4)\)[/tex] on the terminal side of an angle [tex]\(\theta\)[/tex] in standard position, and we need to determine the value of [tex]\(\cos \theta\)[/tex].
### Step-by-Step Solution:
1. Identify the Coordinates:
The point given is [tex]\((3, -4)\)[/tex]. Here, [tex]\(x = 3\)[/tex] and [tex]\(y = -4\)[/tex].
2. Calculate the Hypotenuse (r):
The hypotenuse r can be found using the Pythagorean theorem:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Plugging in the values:
[tex]\[
r = \sqrt{3^2 + (-4)^2}
\][/tex]
[tex]\[
r = \sqrt{9 + 16}
\][/tex]
[tex]\[
r = \sqrt{25}
\][/tex]
[tex]\[
r = 5
\][/tex]
3. Determine [tex]\(\cos \theta\)[/tex]:
[tex]\(\cos \theta\)[/tex] is defined as the ratio of the adjacent side (x) to the hypotenuse (r):
[tex]\[
\cos \theta = \frac{x}{r}
\][/tex]
Using our calculated hypotenuse and given x value:
[tex]\[
\cos \theta = \frac{3}{5}
\][/tex]
So, the value of [tex]\(\cos \theta\)[/tex] is [tex]\(\frac{3}{5}\)[/tex].
### Answer:
[tex]\[
\boxed{\frac{3}{5}}
\][/tex]
Therefore, the correct answer is [tex]\(\text{C.} \frac{3}{5}\)[/tex].
