Find the exact value of \( \csc \theta \), given that \( \cot \theta=-\frac{1}{4} \) and \( \theta \) is in quadrant IV. Rationalize denominators when applicable. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. \( 60 \% \). A. csc \( \theta=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) \( 1.4,67 \) B. The function is undefined.

Given that \( \cot \theta = -\frac{1}{4} \) and \( \theta \) is in Quadrant IV, we can find \( \csc \theta \) as follows: 1. **Understand the Trigonometric Relationships:** - \( \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{x}{y} = -\frac{1}{4} \) - In Quadrant IV, \( x > 0 \) and \( y < 0 \). 2. **Assign Values to \( x \) and \( y \):** Let \( x = 1 \) and \( y = -4 \) to satisfy \( \cot \theta = -\frac{1}{4} \). 3. **Find the Hypotenuse \( r \):** \[ r = \sqrt{x^2 + y^2} = \sqrt{1^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17} \] 4. **Determine \( \sin \theta \) and \( \csc \theta \):** \[ \sin \theta = \frac{y}{r} = \frac{-4}{\sqrt{17}} \] Therefore, \[ \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{17}}{-4} = -\frac{\sqrt{17}}{4} \] 5. **Rationalize the Denominator:** The expression \( -\frac{\sqrt{17}}{4} \) already has a rational denominator. **Final Answer:** **A.** \( \csc \theta = -\dfrac{\sqrt{17}}{4} \)