To find the coordinates of point [tex]\( P(x, y) \)[/tex] on the terminal ray of angle [tex]\( \theta \)[/tex] given that [tex]\(\theta\)[/tex] is between [tex]\(\pi\)[/tex] and [tex]\(\frac{3\pi}{2}\)[/tex] radians, and [tex]\(\csc \theta = -\frac{5}{2}\)[/tex], follow these steps:
1. Determine [tex]\(\sin \theta\)[/tex]:
Since [tex]\(\csc \theta = \frac{1}{\sin \theta}\)[/tex], we have:
[tex]\[
\sin \theta = \frac{1}{\csc \theta} = \frac{1}{-\frac{5}{2}} = -\frac{2}{5}
\][/tex]
2. Find the values for [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
The values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] can be found using the Pythagorean theorem and the relationship between sine and cosine.
From the unit circle:
[tex]\[
\sin \theta = \frac{y}{r}
\][/tex]
Given [tex]\(\sin \theta = -\frac{2}{5}\)[/tex], choose a radius [tex]\( r \)[/tex] for simplicity, [tex]\( r = 5 \)[/tex]:
[tex]\[
y = \sin \theta \times r = -\frac{2}{5} \times 5 = -2
\][/tex]
3. Apply the Pythagorean theorem:
The Pythagorean theorem states that:
[tex]\[
r^2 = x^2 + y^2
\][/tex]
With [tex]\( r = 5 \)[/tex] and [tex]\( y = -2 \)[/tex]:
[tex]\[
5^2 = x^2 + (-2)^2 \implies 25 = x^2 + 4 \implies x^2 = 21 \implies x = \sqrt{21}
\][/tex]
4. Determine the sign of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
Since [tex]\(\theta\)[/tex] is between [tex]\(\pi\)[/tex] and [tex]\(\frac{3\pi}{2}\)[/tex] radians (in the third quadrant), both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] are negative.
So, [tex]\( x = -\sqrt{21} \)[/tex] and [tex]\( y = -2 \)[/tex].
5. Coordinates:
Thus, the coordinates of point [tex]\( P(x, y) \)[/tex] are:
[tex]\[
P(-\sqrt{21}, -2)
\][/tex]
Therefore, the correct answer is:
[tex]\[
P(-\sqrt{21}, -2)
\][/tex]
