$\theta$ is an acute angle and $\sin \theta$ and $\cos \theta$ are given. Use identities to find $\tan \theta, \csc \theta, \sec \theta$, and $\cot \theta$. Where necessary, rationalize denominators. \[ \sin \theta=\frac{12}{13}, \cos \theta=\frac{5}{13} \] \[ \tan \theta=\square \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) \[ \csc \theta=\square \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) \[ \sec \theta=\square \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.) \[ \cot \theta=\square \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Rationalize all denominators.)
Solution
Step 1 ：\(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
Step 2 ：Substitute \(\sin \theta = \frac{12}{13}\) and \(\cos \theta = \frac{5}{13}\) into the equation
Step 3 ：\(\tan \theta = \frac{\frac{12}{13}}{\frac{5}{13}} = \frac{12}{5}\)
Step 4 ：\(\boxed{\tan \theta = \frac{12}{5}}\)
Step 5 ：\(\csc \theta = \frac{1}{\sin \theta}\)
Step 6 ：Substitute \(\sin \theta = \frac{12}{13}\) into the equation
Step 7 ：\(\csc \theta = \frac{1}{\frac{12}{13}} = \frac{13}{12}\)
Step 8 ：\(\boxed{\csc \theta = \frac{13}{12}}\)
Step 9 ：\(\sec \theta = \frac{1}{\cos \theta}\)
Step 10 ：Substitute \(\cos \theta = \frac{5}{13}\) into the equation
Step 11 ：\(\sec \theta = \frac{1}{\frac{5}{13}} = \frac{13}{5}\)
Step 12 ：\(\boxed{\sec \theta = \frac{13}{5}}\)
Step 13 ：\(\cot \theta = \frac{1}{\tan \theta}\)
Step 14 ：Substitute \(\tan \theta = \frac{12}{5}\) into the equation
Step 15 ：\(\cot \theta = \frac{1}{\frac{12}{5}} = \frac{5}{12}\)
Step 16 ：\(\boxed{\cot \theta = \frac{5}{12}}\)
