Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \tan \theta=-3 \) (Simplity your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \csc \theta=-\frac{\sqrt{10}}{3} \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sec \theta= \) \( \square \) (Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed. Rationalize all denominators.) B. The function is undefined.

It appears that you've provided multiple parts of a problem related to trigonometric functions, but some essential information or context seems to be missing. To accurately assist you, I need more details about the specific conditions or the scenario in which these trigonometric functions are being evaluated. However, I can offer some general guidance on how to approach such problems: ### Understanding When Trigonometric Functions Are Defined or Undefined 1. **Tangent Function (\( \tan \theta \))**: - **Defined When**: \( \cos \theta \neq 0 \) - **Undefined When**: \( \cos \theta = 0 \) - **Given \( \tan \theta = -3 \)**: You can find \( \sin \theta \) and \( \cos \theta \) using the identity \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). For example, if \( \tan \theta = -3 \), you might set up a right triangle where the opposite side is -3 and the adjacent side is 1, then find the hypotenuse using the Pythagorean theorem. 2. **Cosecant Function (\( \csc \theta \))**: - **Defined When**: \( \sin \theta \neq 0 \) - **Undefined When**: \( \sin \theta = 0 \) - **Given \( \csc \theta = -\frac{\sqrt{10}}{3} \)**: Since \( \csc \theta = \frac{1}{\sin \theta} \), you can find \( \sin \theta \) by taking the reciprocal, i.e., \( \sin \theta = -\frac{3}{\sqrt{10}} \). It's recommended to rationalize the denominator: \( \sin \theta = -\frac{3\sqrt{10}}{10} \). 3. **Secant Function (\( \sec \theta \))**: - **Defined When**: \( \cos \theta \neq 0 \) - **Undefined When**: \( \cos \theta = 0 \) - **Given \( \sec \theta = \) [ ]**: Without additional information (like \( \cos \theta \) or another trigonometric function value), it's impossible to determine the exact value of \( \sec \theta \). If \( \cos \theta \) were known, \( \sec \theta \) could be found as its reciprocal. ### Next Steps To provide precise answers: - **Provide the Complete Problem Statement**: Include any given values, diagrams (if any), and specific questions you're trying to answer. - **Specify What You Need Help With**: Are you looking to simplify expressions, find exact values, determine definitions, or something else? Feel free to share the complete details, and I'll be more than happy to help you work through the problem step by step!