Find the exact values of the remaining trigonometric functions of $\theta$ satisfying the given conditions. (If an answer is undefined, enter UNDEFINED.) \[ \begin{array}{l} \cos \theta=\frac{15}{17}, \quad \tan \theta<0 \\ \sin \theta=\square \\ \tan \theta=\square \\ \csc \theta=\square \\ \sec \theta=\square \\ \cot \theta=\square \end{array} \] Submit Answer
Solution
Step 1 ：We are given that \(\cos \theta = \frac{15}{17}\) and \(\tan \theta < 0\). This implies that \(\theta\) is in the fourth quadrant, where cosine is positive and tangent is negative.
Step 2 ：We can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\). Since \(\theta\) is in the fourth quadrant, \(\sin \theta\) should be negative. Solving for \(\sin \theta\), we get \(\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{15}{17}\right)^2} = -\frac{8}{17}\).
Step 3 ：Next, we can find \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-8/17}{15/17} = -\frac{8}{15}\).
Step 4 ：We can find \(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{-8/17} = -\frac{17}{8}\).
Step 5 ：We can find \(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{15/17} = \frac{17}{15}\).
Step 6 ：We can find \(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{-8/15} = -\frac{15}{8}\).
Step 7 ：Final Answer: \(\sin \theta= \boxed{-\frac{8}{17}}, \tan \theta= \boxed{-\frac{8}{15}}, \csc \theta= \boxed{-\frac{17}{8}}, \sec \theta= \boxed{\frac{17}{15}}, \cot \theta= \boxed{-\frac{15}{8}}\)
