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.
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To find the exact value of \( \csc \theta \), we start from the information given: \( \cot \theta = -\frac{1}{4} \) and \( \theta \) is in quadrant IV. Since \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), we can set \( \cos \theta = -1 \) (for convenience) and \( \sin \theta = 4 \) because \( \cot \theta \) is negative in the fourth quadrant. Now, using the Pythagorean identity, we find \( r \): \[ r = \sqrt{(\cos \theta)^2 + (\sin \theta)^2} = \sqrt{(-1)^2 + (4)^2} = \sqrt{1 + 16} = \sqrt{17} \] Thus, we determine \( \sin \theta \) and \( \cos \theta \): \[ \sin \theta = \frac{4}{\sqrt{17}}, \quad \cos \theta = \frac{-1}{\sqrt{17}}. \] Now, to find \( \csc \theta \), which is the reciprocal of \( \sin \theta \): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{17}}{4}. \] So the answer is: A. csc \( \theta = \frac{\sqrt{17}}{4} \).