Part 5 of 5 Points: 0 of 1 Find the exact value of each of the remaining trigonometric functions of \( \theta \). Rationalize denominators when applicable. \( \cot \theta=-\frac{\sqrt{3}}{7} \), given that \( \theta \) is in quadrant II B. The function is undefined. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \boldsymbol{\operatorname { t a n }} \theta=-\frac{7 \sqrt{3}}{3} \) (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. \( \csc \theta=\frac{2 \sqrt{13}}{7} \) (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. \( \sec \theta= \) \( \square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.) B. The function is undefined. View an example Get more help -

To find the exact values of the remaining trigonometric functions of \( \theta \) given that \( \cot \theta = -\frac{\sqrt{3}}{7} \) and \( \theta \) is in **Quadrant II**, we'll follow these steps: 1. **Determine the Signs in Quadrant II:** - **Sine (\( \sin \theta \))**: Positive - **Cosine (\( \cos \theta \))**: Negative - **Tangent (\( \tan \theta \))**: Negative - **Cosecant (\( \csc \theta \))**: Positive - **Secant (\( \sec \theta \))**: Negative - **Cotangent (\( \cot \theta \))**: Negative (Given) 2. **Use the Definition of Cotangent:** \[ \cot \theta = \frac{\text{Adjacent}}{\text{Opposite}} = -\frac{\sqrt{3}}{7} \] - **Adjacent Side**: \( -\sqrt{3} \) (negative in Quadrant II) - **Opposite Side**: \( 7 \) (positive in Quadrant II) 3. **Find the Hypotenuse Using the Pythagorean Theorem:** \[ \text{Hypotenuse} = \sqrt{(\text{Adjacent})^2 + (\text{Opposite})^2} = \sqrt{(-\sqrt{3})^2 + 7^2} = \sqrt{3 + 49} = \sqrt{52} = 2\sqrt{13} \] 4. **Calculate the Remaining Trigonometric Functions:** - **Sine (\( \sin \theta \)):** \[ \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{7}{2\sqrt{13}} = \frac{7\sqrt{13}}{26} \] - **Cosine (\( \cos \theta \)):** \[ \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{-\sqrt{3}}{2\sqrt{13}} = \frac{-\sqrt{39}}{26} \] - **Tangent (\( \tan \theta \)):** \[ \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{7}{-\sqrt{3}} = -\frac{7\sqrt{3}}{3} \] - **Cosecant (\( \csc \theta \)):** \[ \csc \theta = \frac{1}{\sin \theta} = \frac{2\sqrt{13}}{7} \] - **Secant (\( \sec \theta \)):** \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{\sqrt{39}}{26}} = -\frac{26}{\sqrt{39}} = -\frac{2\sqrt{39}}{3} \] - **Cotangent (\( \cot \theta \)):** \[ \cot \theta = -\frac{\sqrt{3}}{7} \quad \text{(Given)} \] 5. **Summary of All Trigonometric Functions:** \[ \begin{aligned} \sin \theta &= \frac{7\sqrt{13}}{26} \\ \cos \theta &= \frac{-\sqrt{39}}{26} \\ \tan \theta &= -\frac{7\sqrt{3}}{3} \\ \csc \theta &= \frac{2\sqrt{13}}{7} \\ \sec \theta &= -\frac{2\sqrt{39}}{3} \\ \cot \theta &= -\frac{\sqrt{3}}{7} \quad \text{(Given)} \end{aligned} \] 6. **Addressing the Multiple Choice Options:** - **For \( \tan \theta \):** - **A.** \( \tan \theta = -\frac{7 \sqrt{3}}{3} \) **✔️ Correct** - **B.** The function is undefined. ❌ Incorrect - **For \( \csc \theta \):** - **A.** \( \csc \theta = \frac{2 \sqrt{13}}{7} \) **✔️ Correct** - **B.** The function is undefined. ❌ Incorrect - **For \( \sec \theta \):** - **A.** \( \sec \theta = -\frac{2\sqrt{39}}{3} \) - **B.** The function is undefined. **Choose A** and fill in the answer: \[ \sec \theta = -\frac{2\sqrt{39}}{3} \] Ensure that all radicals are simplified and denominators are rationalized as shown above.