If $$ \sin heta=-\frac{2}{3} $$, which of the following are possible? Choose all correct answers. $$ \sec heta=\frac{3}{\sqrt{5}} $$ and $$ an heta=-\frac{2}{\sqrt{5}} $$ $$ \sec heta=-\frac{3}{2} $$ and $$ an heta=\frac{2}{\sqrt{5}} $$ $$ \cos heta=-\frac{\sqrt{5}}{3} $$ and $$ an heta=\frac{2}{\sqrt{5}} $$ $$ \cos heta=\frac{\sqrt{5}}{3} $$ and $$ an heta=\frac{2}{\sqrt{5}} $$

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If $$ \sin 	heta=-\frac{2}{3} $$, which of the following are possible? Choose all correct answers. $$ \sec 	heta=\frac{3}{\sqrt{5}} $$ and $$ 	an 	heta=-\frac{2}{\sqrt{5}} $$ $$ \sec 	heta=-\frac{3}{2} $$ and $$ 	an 	heta=\frac{2}{\sqrt{5}} $$ $$ \cos 	heta=-\frac{\sqrt{5}}{3} $$ and $$ 	an 	heta=\frac{2}{\sqrt{5}} $$ $$ \cos 	heta=\frac{\sqrt{5}}{3} $$ and $$ 	an 	heta=\frac{2}{\sqrt{5}} $$

UpStudy AI · Accepted Answer

Given $$ \sin \theta = -\frac{2}{3} $$, we can determine the possible values for $$ \cos \theta $$ and $$ \tan \theta $$.

1. **Determine $$ \cos \theta $$:**
   $$
   \sin^2 \theta + \cos^2 \theta = 1 \
   \left(-\frac{2}{3}\right)^2 + \cos^2 \theta = 1 \
   \frac{4}{9} + \cos^2 \theta = 1 \
   \cos^2 \theta = \frac{5}{9} \
   \cos \theta = \pm \frac{\sqrt{5}}{3}
   $$

2. **Determine $$ \tan \theta $$:**
   $$
   \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{2}{3}}{\pm \frac{\sqrt{5}}{3}} = \mp \frac{2}{\sqrt{5}}
   $$

3. **Determine $$ \sec \theta $$:**
   $$
   \sec \theta = \frac{1}{\cos \theta} = \frac{3}{\pm \sqrt{5}}
   $$

Now, evaluate each option:

1. **$$ \sec \theta = \frac{3}{\sqrt{5}} $$ and $$ \tan \theta = -\frac{2}{\sqrt{5}} $$:**
   - This corresponds to $$ \cos \theta = \frac{\sqrt{5}}{3} $$ and $$ \tan \theta = -\frac{2}{\sqrt{5}} $$.
   - This is consistent with $$ \sin \theta = -\frac{2}{3} $$ in Quadrant IV.

2. **$$ \sec \theta = -\frac{3}{2} $$ and $$ \tan \theta = \frac{2}{\sqrt{5}} $$:**
   - This would imply $$ \cos \theta = -\frac{2}{3} $$.
   - However, this violates the Pythagorean identity since $$ \sin^2 \theta + \cos^2 \theta \neq 1 $$.
   - **Invalid.**

3. **$$ \cos \theta = -\frac{\sqrt{5}}{3} $$ and $$ \tan \theta = \frac{2}{\sqrt{5}} $$:**
   - This corresponds to $$ \tan \theta = \frac{2}{\sqrt{5}} $$.
   - This is consistent with $$ \sin \theta = -\frac{2}{3} $$ in Quadrant III.

4. **$$ \cos \theta = \frac{\sqrt{5}}{3} $$ and $$ \tan \theta = \frac{2}{\sqrt{5}} $$:**
   - Given $$ \sin \theta = -\frac{2}{3} $$, this would imply $$ \tan \theta = -\frac{2}{\sqrt{5}} $$.
   - **Inconsistent with the given $$ \tan \theta $$.**

**Therefore, the correct possible options are the first and third statements.**

**Answer:** All except the second and fourth choices are possible.
The possible answers are: - $$ \sec \theta = \frac{3}{\sqrt{5}} $$ and $$ \tan \theta = -\frac{2}{\sqrt{5}} $$ - $$ \cos \theta = -\frac{\sqrt{5}}{3} $$ and $$ \tan \theta = \frac{2}{\sqrt{5}} $$ These correspond to options A and C.

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