Find the remaining five trigonometic functions of $\theta$ \[ \cot \theta=\frac{4}{3}, \sin \theta>0 \] Complete the following table. \[ \begin{array}{ll} \sin \theta=\square & \csc \theta=\square \\ \cos \theta=\square & \sec \theta=\square \\ \tan \theta=\square & \cot \theta=\frac{4}{3} \end{array} \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)
Solution
Step 1 ：We know that \(\cot \theta=\frac{4}{3}\), which is the reciprocal of \(\tan \theta\). So, \(\tan \theta=\frac{3}{4}\).
Step 2 ：We also know that \(\tan \theta=\frac{\sin \theta}{\cos \theta}\). So, we can use the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\) to find \(\sin \theta\) and \(\cos \theta\).
Step 3 ：Since \(\sin \theta>0\), we are in the first or second quadrant where both \(\sin \theta\) and \(\cos \theta\) are positive.
Step 4 ：Finally, we can find \(\csc \theta\) and \(\sec \theta\) which are the reciprocals of \(\sin \theta\) and \(\cos \theta\) respectively.
Step 5 ：Final Answer: \[\begin{array}{ll} \sin \theta=\boxed{0.6} & \csc \theta=\boxed{1.67} \\ \cos \theta=\boxed{0.8} & \sec \theta=\boxed{1.25} \\ \tan \theta=\boxed{0.75} & \cot \theta=\boxed{1.33} \end{array}\]
