Find the value of each of the six trigonometric functions of the angle $\theta$ in the figure. $\sqrt{29}$ \[ \sin \theta=\frac{2}{\sqrt{29}} \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) \[ \cos \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) $\tan \theta=$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) \[ \csc \theta= \] (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) \[ \sec \theta= \] (Simplify your-answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.) $\cot \theta=$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression. Type an exact answer, using radicals as needed.)
Solution
Step 1 ：Given that the hypotenuse of the triangle is \(\sqrt{29}\) and the opposite side to the angle \(\theta\) is 2, we can use the Pythagorean theorem to find the adjacent side to the angle \(\theta\).
Step 2 ：Using the Pythagorean theorem, we find that the adjacent side is 5.
Step 3 ：We can now use the definitions of the six trigonometric functions to find their values.
Step 4 ：\(\sin \theta = \frac{opposite}{hypotenuse} = \frac{2}{\sqrt{29}}\)
Step 5 ：\(\cos \theta = \frac{adjacent}{hypotenuse} = \frac{5}{\sqrt{29}}\)
Step 6 ：\(\tan \theta = \frac{opposite}{adjacent} = \frac{2}{5}\)
Step 7 ：\(\csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{29}}{2}\)
Step 8 ：\(\sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{29}}{5}\)
Step 9 ：\(\cot \theta = \frac{1}{\tan \theta} = \frac{5}{2}\)
Step 10 ：Final Answer: \(\boxed{\sin \theta = \frac{2}{\sqrt{29}}, \cos \theta = \frac{5}{\sqrt{29}}, \tan \theta = \frac{2}{5}, \csc \theta = \frac{\sqrt{29}}{2}, \sec \theta = \frac{\sqrt{29}}{5}, \cot \theta = \frac{5}{2}}\)
