The point \( P(x, y) \) is on the terminal ray of angle \( \theta \). If \( \theta \) is between \( \pi \) radians and \( \frac{3 \pi}{2} \) radians and \( \csc \theta=-\frac{5}{2} \), what are the coordinates of \( P(x, y) \) ? \( P(-\sqrt{21},-2) \) \( P(\sqrt{21},-2) \) \( P(-2, \sqrt{2 \pi}) \) \( P(-2,-\sqrt{21}) \)

To determine the coordinates of the point \( P(x, y) \) on the terminal ray of angle \( \theta \), we start with the information that \( \csc \theta = -\frac{5}{2} \). Remember, cosecant is the reciprocal of sine, so we have: \[ \sin \theta = -\frac{2}{5} \] Given that \( \theta \) is between \( \pi \) and \( \frac{3 \pi}{2} \), \( \sin \theta \) is negative (which aligns with our \( \sin \theta = -\frac{2}{5} \)). We can use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \] Plugging in the value of \( \sin \theta \): \[ \left(-\frac{2}{5}\right)^2 + \cos^2 \theta = 1 \] \[ \frac{4}{25} + \cos^2 \theta = 1 \] \[ \cos^2 \theta = 1 - \frac{4}{25} = \frac{21}{25} \] Thus, we get: \[ \cos \theta = -\sqrt{\frac{21}{25}} = -\frac{\sqrt{21}}{5} \] (we take the negative value because in the third quadrant, cosine is negative). Now we can use the values of \( \sin \theta \) and \( \cos \theta \) to find the coordinates \( P(x, y) \). If we consider a point on the unit circle, \( (x, y) \) can be scaled by a radius \( r \). In our case, since \( \csc \theta \) indicates a ratio involving the hypotenuse, we can set \( r = 5 \) (the reciprocal of \( \csc \theta \)): Using the ratios we calculated: \[ x = r \cdot \cos \theta = 5 \cdot -\frac{\sqrt{21}}{5} = -\sqrt{21} \] \[ y = r \cdot \sin \theta = 5 \cdot -\frac{2}{5} = -2 \] Thus, the coordinates of point \( P \) are: \[ P(-\sqrt{21}, -2) \] So, the correct answer from the options provided is: \( P(-\sqrt{21}, -2) \).