Let $\theta$ be an angle such that $\csc \theta=\frac{5}{3}$ and $\tan \theta>0$. Find the exact values of $\cot \theta$ and $\cos \theta$. \[ \begin{array}{l} \cot \theta=\prod \\ \cos \theta= \end{array} \]
Solution
Step 1 ：Let \(\theta\) be an angle such that \(\csc \theta=\frac{5}{3}\) and \(\tan \theta>0\). We are to find the exact values of \(\cot \theta\) and \(\cos \theta\).
Step 2 ：The cosecant of an angle is defined as the reciprocal of the sine of the angle. So, if \(\csc \theta=\frac{5}{3}\), then \(\sin \theta=\frac{3}{5}\).
Step 3 ：The tangent of an angle is positive in the first and third quadrants. Since \(\tan \theta>0\) and \(\sin \theta>0\), we know that \(\theta\) is in the first quadrant.
Step 4 ：The cotangent of an angle is the reciprocal of the tangent of the angle. We can find the tangent of the angle using the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). Solving for \(\cos \theta\), we get \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). Then, we can find \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).
Step 5 ：Substituting the values we have, \(\sin \theta = 0.6\) and \(\cos \theta = 0.8\), we find that \(\cot \theta = 1.3333333333333335\).
Step 6 ：Final Answer: \(\cot \theta=\boxed{1.3333333333333335}\) and \(\cos \theta=\boxed{0.8}\)
