To determine the coordinates of the point [tex]\( P(x, y) \)[/tex] that lies on the terminal ray of the angle [tex]\( \theta \)[/tex], given that [tex]\( \theta \)[/tex] is between [tex]\( \pi \)[/tex] and [tex]\( \frac{3\pi}{2} \)[/tex] and [tex]\( \csc \theta = -\frac{5}{2} \)[/tex], we can analyze the following steps:
1. Understanding the given information:
- [tex]\( \csc \theta = -\frac{5}{2} \)[/tex].
- The cosecant function, [tex]\( \csc \theta \)[/tex], is the reciprocal of the sine function, [tex]\( \sin \theta \)[/tex]. Therefore,
[tex]\[
\csc \theta = \frac{1}{\sin \theta}
\][/tex]
- Given [tex]\( \csc \theta = -\frac{5}{2} \)[/tex], we can write:
[tex]\[
\sin \theta = -\frac{2}{5}
\][/tex]
2. Finding [tex]\( \cos \theta \)[/tex] using Pythagorean identity:
- The Pythagorean identity states:
[tex]\[
\sin^2 \theta + \cos^2 \theta = 1
\][/tex]
- Substitute [tex]\( \sin \theta = -\frac{2}{5} \)[/tex]:
[tex]\[
\left( -\frac{2}{5} \right)^2 + \cos^2 \theta = 1
\][/tex]
[tex]\[
\frac{4}{25} + \cos^2 \theta = 1
\][/tex]
[tex]\[
\cos^2 \theta = 1 - \frac{4}{25}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{25}{25} - \frac{4}{25}
\][/tex]
[tex]\[
\cos^2 \theta = \frac{21}{25}
\][/tex]
[tex]\[
\cos \theta = \pm \sqrt{\frac{21}{25}} = \pm \frac{\sqrt{21}}{5}
\][/tex]
3. Determining the sign of [tex]\( \cos \theta \)[/tex]:
- Since [tex]\( \theta \)[/tex] is between [tex]\( \pi \)[/tex] and [tex]\( \frac{3\pi}{2} \)[/tex], it is in the third quadrant where both sine and cosine values are negative. Therefore, we have:
[tex]\[
\cos \theta = -\frac{\sqrt{21}}{5}
\][/tex]
4. Finding the coordinates:
- Any point [tex]\( P \)[/tex] on the terminal ray of [tex]\( \theta \)[/tex] can be represented as [tex]\( (r \cos \theta, r \sin \theta) \)[/tex], where [tex]\( r \)[/tex] is the hypotenuse of the triangle formed in the unit circle, which is given to be 5.
- So we have:
[tex]\[
x = r \cos \theta = 5 \left( -\frac{\sqrt{21}}{5} \right) = -\sqrt{21}
\][/tex]
[tex]\[
y = r \sin \theta = 5 \left( -\frac{2}{5} \right) = -2
\][/tex]
Therefore, the coordinates of [tex]\( P(x, y) \)[/tex] are:
[tex]\[
P( -\sqrt{21}, -2 )
\][/tex]
From the given choices, the correct answer is:
[tex]\[ P( -\sqrt{21}, -2 ) \][/tex]
