Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sin \theta=\frac{2 \sqrt{6}}{5} \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \cos \theta=-\frac{1}{5} \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) Belect the correct choice below and, if necessary, fill in the answer box to complete your choice. A. tan \( \theta=-2 \sqrt{6} \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. csc \( \theta= \) A \( ^{-15} \). (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined.

Let's evaluate each trigonometric function based on the given choices. We'll assume that there's a right triangle involved where \( \cos \theta = -\frac{1}{5} \). This implies that \( \theta \) is in either the second or third quadrant where cosine is negative. ### 1. \( \sin \theta \) **Choices:** - **A.** \( \sin \theta = \frac{2 \sqrt{6}}{5} \) - **B.** The function is undefined. **Evaluation:** Using the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1 \\ \sin^2 \theta = 1 - \cos^2 \theta = 1 - \left(-\frac{1}{5}\right)^2 = 1 - \frac{1}{25} = \frac{24}{25} \\ \sin \theta = \pm \frac{\sqrt{24}}{5} = \pm \frac{2\sqrt{6}}{5} \] Since \( \cos \theta \) is negative, if \( \theta \) is in the second quadrant, \( \sin \theta \) is positive. If in the third quadrant, it's negative. Given no additional information about the quadrant, we'll consider the positive value. **Correct Choice:** **A.** \( \sin \theta = \frac{2 \sqrt{6}}{5} \) --- ### 2. \( \cos \theta \) **Choices:** - **A.** \( \cos \theta = -\frac{1}{5} \) - **B.** The function is undefined. **Evaluation:** The value \( \cos \theta = -\frac{1}{5} \) is defined based on the given information. **Correct Choice:** **A.** \( \cos \theta = -\frac{1}{5} \) --- ### 3. \( \tan \theta \) **Choices:** - **A.** \( \tan \theta = -2 \sqrt{6} \) - **B.** The function is undefined. **Evaluation:** \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{6}}{5}}{-\frac{1}{5}} = -2\sqrt{6} \] **Correct Choice:** **A.** \( \tan \theta = -2 \sqrt{6} \) --- ### 4. \( \csc \theta \) **Choices:** - **A.** \( \csc \theta = \frac{5 \sqrt{6}}{12} \) *(Assuming the intended expression was meant to be completed correctly)* - **B.** The function is undefined. **Evaluation:** \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{2\sqrt{6}}{5}} = \frac{5}{2\sqrt{6}} = \frac{5\sqrt{6}}{12} \] **Correct Choice:** **A.** \( \csc \theta = \frac{5 \sqrt{6}}{12} \) --- **Summary of Selections:** 1. **A.** \( \sin \theta = \frac{2 \sqrt{6}}{5} \) 2. **A.** \( \cos \theta = -\frac{1}{5} \) 3. **A.** \( \tan \theta = -2 \sqrt{6} \) 4. **A.** \( \csc \theta = \frac{5 \sqrt{6}}{12} \) All functions are **defined**, and their corresponding expressions are simplified as required.