This questions 1 pomt(9) possible Find the exact value of each of the remaining trigonometric functions of 0 . Rationalize denominators when applicable, \( \tan \theta=-\frac{3}{4} \), given that 0 is in quadrant IV Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \sin \theta= \) \( \square \) (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. \( \cos \theta= \) \( \square \) (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. \( \csc 0= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

We are given that \[ \tan \theta = -\frac{3}{4} \] and that \(\theta\) is in quadrant IV. In quadrant IV, the cosine is positive and the sine is negative. Step 1. Express \(\sin \theta\) and \(\cos \theta\) in terms of a common factor: \[ \sin \theta = -3k \quad \text{and} \quad \cos \theta = 4k \] so that \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-3k}{4k} = -\frac{3}{4}. \] Step 2. Use the Pythagorean identity: \[ \sin^2 \theta + \cos^2 \theta = 1. \] Substitute the expressions for \(\sin \theta\) and \(\cos \theta\): \[ (-3k)^2 + (4k)^2 = 9k^2 + 16k^2 = 25k^2 = 1. \] Solve for \(k\): \[ 25k^2 = 1 \quad \Rightarrow \quad k^2 = \frac{1}{25} \quad \Rightarrow \quad k = \frac{1}{5}. \] Step 3. Find the values of \(\sin \theta\) and \(\cos \theta\): \[ \sin \theta = -3k = -\frac{3}{5} \] \[ \cos \theta = 4k = \frac{4}{5} \] Step 4. Find the reciprocal function \(\csc \theta\): \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{3}{5}} = -\frac{5}{3}. \] Thus, the answers are: A. \(\sin \theta = -\frac{3}{5}\) A. \(\cos \theta = \frac{4}{5}\) A. \(\csc \theta = -\frac{5}{3}\)