To find the exact value of [tex]\(\sin \theta\)[/tex] where the terminal arm of the angle in standard position passes through the point [tex]\((-3, 4)\)[/tex], we can follow these steps:
1. Identify the coordinates:
- We are given the point [tex]\((-3, 4)\)[/tex]. Here, [tex]\( x = -3 \)[/tex] and [tex]\( y = 4 \)[/tex].
2. Calculate the distance [tex]\( r \)[/tex]:
- The distance from the origin to the point [tex]\((-3, 4)\)[/tex] is the hypotenuse of a right triangle with legs of lengths [tex]\(|x|\)[/tex] and [tex]\(|y|\)[/tex]. The formula for the hypotenuse [tex]\( r \)[/tex] is:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
- Substitute 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]\(\sin \theta\)[/tex]:
- The sine of the angle [tex]\(\theta\)[/tex] is given by:
[tex]\[
\sin \theta = \frac{y}{r}
\][/tex]
- Substitute the values:
[tex]\[
\sin \theta = \frac{4}{5}
\][/tex]
Thus, the exact value of [tex]\(\sin \theta\)[/tex] is:
C. [tex]\(\frac{4}{5}\)[/tex]
