Find the exact value of each of the remaining trigonometric functions of \( \theta \). \[ \sec \theta=17, \tan \theta>0 \]

We are given \[ \sec \theta = 17 \quad \text{and} \quad \tan \theta > 0. \] Since \[ \sec \theta = \frac{1}{\cos \theta}, \] we have \[ \cos \theta = \frac{1}{17}. \] Because \(\tan \theta > 0\), both \(\sin \theta\) and \(\cos \theta\) must have the same sign. Since \(\cos \theta = \frac{1}{17} > 0\), it follows that \[ \sin \theta > 0. \] Thus, \(\theta\) is in the first quadrant. ### Step 1. Find \(\sin \theta\) Using the Pythagorean identity, \[ \sin^2 \theta + \cos^2 \theta = 1, \] we substitute \(\cos \theta = \frac{1}{17}\): \[ \sin^2 \theta + \left(\frac{1}{17}\right)^2 = 1. \] This gives \[ \sin^2 \theta + \frac{1}{289} = 1. \] Subtract \(\frac{1}{289}\) from both sides: \[ \sin^2 \theta = 1 - \frac{1}{289} = \frac{289}{289} - \frac{1}{289} = \frac{288}{289}. \] Taking the positive square root (since \(\sin \theta > 0\)): \[ \sin \theta = \sqrt{\frac{288}{289}} = \frac{\sqrt{288}}{17}. \] Simplify \(\sqrt{288}\): \[ \sqrt{288} = \sqrt{144 \cdot 2} = 12\sqrt{2}. \] Thus, \[ \sin \theta = \frac{12\sqrt{2}}{17}. \] ### Step 2. Find \(\tan \theta\) By definition, \[ \tan \theta = \frac{\sin \theta}{\cos \theta}. \] Substitute the values: \[ \tan \theta = \frac{\frac{12\sqrt{2}}{17}}{\frac{1}{17}} = 12\sqrt{2}. \] ### Step 3. Find the remaining trigonometric functions 1. **Cosecant (\(\csc \theta\))** \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{12\sqrt{2}}{17}} = \frac{17}{12\sqrt{2}}. \] Rationalize the denominator: \[ \csc \theta = \frac{17\sqrt{2}}{24}. \] 2. **Cotangent (\(\cot \theta\))** \[ \cot \theta = \frac{\cos \theta}{\sin \theta} = \frac{\frac{1}{17}}{\frac{12\sqrt{2}}{17}} = \frac{1}{12\sqrt{2}}. \] Rationalize the denominator: \[ \cot \theta = \frac{\sqrt{2}}{24}. \] 3. **Secant (\(\sec \theta\))** is given by: \[ \sec \theta = 17. \] 4. **Cosine (\(\cos \theta\))** is: \[ \cos \theta = \frac{1}{17}. \] ### Final Values \[ \begin{aligned} \cos \theta &= \frac{1}{17}, \\ \sin \theta &= \frac{12\sqrt{2}}{17}, \\ \tan \theta &= 12\sqrt{2}, \\ \sec \theta &= 17, \\ \csc \theta &= \frac{17\sqrt{2}}{24}, \\ \cot \theta &= \frac{\sqrt{2}}{24}. \end{aligned} \]