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

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

UpStudy AI · Accepted Answer

To find the coordinates of the point $$ P(x, y) $$ on the terminal ray of angle $$ \theta $$ given that $$ \csc \theta = -\frac{5}{2} $$, we can follow these steps:

1. **Understand the relationship between cosecant and sine**:
   $$
   \csc \theta = \frac{1}{\sin \theta}
   $$
   Therefore, if $$ \csc \theta = -\frac{5}{2} $$, then:
   $$
   \sin \theta = -\frac{2}{5}
   $$

2. **Determine the quadrant**:
   Since $$ \theta $$ is between $$ \pi $$ and $$ \frac{3\pi}{2} $$, it is in the third quadrant where both sine and cosine are negative.

3. **Use the Pythagorean identity**:
   We know that:
   $$
   \sin^2 \theta + \cos^2 \theta = 1
   $$
   Substituting $$ \sin \theta $$:
   $$
   \left(-\frac{2}{5}\right)^2 + \cos^2 \theta = 1
   $$
   This simplifies to:
   $$
   \frac{4}{25} + \cos^2 \theta = 1
   $$
   $$
   \cos^2 \theta = 1 - \frac{4}{25} = \frac{25}{25} - \frac{4}{25} = \frac{21}{25}
   $$
   Taking the square root gives:
   $$
   \cos \theta = -\sqrt{\frac{21}{25}} = -\frac{\sqrt{21}}{5}
   $$
   (negative because cosine is negative in the third quadrant).

4. **Find the coordinates**:
   The coordinates $$ P(x, y) $$ can be expressed in terms of the radius $$ r $$:
   $$
   x = r \cos \theta, \quad y = r \sin \theta
   $$
   We can choose $$ r = 5 $$ (the hypotenuse) for simplicity:
   $$
   x = 5 \left(-\frac{\sqrt{21}}{5}\right) = -\sqrt{21}
   $$
   $$
   y = 5 \left(-\frac{2}{5}\right) = -2
   $$

Thus, the coordinates of point $$ P $$ are $$ P(-\sqrt{21}, -2) $$.

The correct answer is:
$$ P(-\sqrt{21}, -2) $$
The coordinates of point $$ P $$ are $$ (-\sqrt{21}, -2) $$.

The point is on the terminal ray of angle . If is between radians and radians and , what are the
coordinates of ?

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The point is on the terminal ray of angle . If is between radians and radians and , what are the coordinates of ?