The exact value of [tex]\sin \theta[/tex], if the terminal arm of the angle in standard position passes through the point [tex](-3, 4)[/tex], is:

A. [tex]-\frac{4}{3}[/tex]
B. [tex]-\frac{3}{5}[/tex]
C. [tex]\frac{4}{5}[/tex]
D. [tex]\frac{5}{4}[/tex]



Answer :

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]