To determine the value of [tex]\(\sin \theta\)[/tex] given that the terminal side of the angle passes through the point [tex]\(P(15, -8)\)[/tex], follow these steps:
1. Identify the coordinates: The point [tex]\(P\)[/tex] is given as [tex]\((15, -8)\)[/tex].
2. Calculate the hypotenuse (r): The hypotenuse [tex]\(r\)[/tex] in this context is the distance from the origin to the point [tex]\(P(15, -8)\)[/tex]. This can be calculated using the distance formula:
[tex]\[
r = \sqrt{x^2 + y^2}
\][/tex]
Here, [tex]\(x = 15\)[/tex] and [tex]\(y = -8\)[/tex].
Substituting these values in:
[tex]\[
r = \sqrt{15^2 + (-8)^2} = \sqrt{225 + 64} = \sqrt{289} = 17
\][/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]
Using [tex]\(y = -8\)[/tex] and [tex]\(r = 17\)[/tex]:
[tex]\[
\sin \theta = \frac{-8}{17}
\][/tex]
From the steps above, we find that:
[tex]\[
\sin \theta = -\frac{8}{17}
\][/tex]
Thus, the correct answer is:
[tex]\[
\sin \theta = -\frac{8}{17}
\][/tex]
