the value of one trigonometric function of an acute angle 3 , find the volues of the ramaining five trigonometric functions of . Gue the eract answar as a olfied fraction, if necessary, Rationalice denominators if necescary. \[ \cos 5=\frac{\sqrt{5}}{5} \]

Given that \(\cos \theta = \frac{\sqrt{5}}{5}\) for an acute angle \(\theta\), we can find the remaining five trigonometric functions as follows: 1. **Sine (\(\sin \theta\))**: \[ \sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{\sqrt{5}}{5}\right)^2} = \sqrt{1 - \frac{5}{25}} = \sqrt{\frac{20}{25}} = \frac{2\sqrt{5}}{5} \] 2. **Tangent (\(\tan \theta\))**: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{2\sqrt{5}}{5}}{\frac{\sqrt{5}}{5}} = 2 \] 3. **Cotangent (\(\cot \theta\))**: \[ \cot \theta = \frac{1}{\tan \theta} = \frac{1}{2} \] 4. **Secant (\(\sec \theta\))**: \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{\sqrt{5}}{5}} = \frac{5}{\sqrt{5}} = \sqrt{5} \] 5. **Cosecant (\(\csc \theta\))**: \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{\frac{2\sqrt{5}}{5}} = \frac{5}{2\sqrt{5}} = \frac{\sqrt{5}}{2} \] **Summary of All Six Trigonometric Functions:** \[ \begin{aligned} \sin \theta &= \frac{2\sqrt{5}}{5} \\ \cos \theta &= \frac{\sqrt{5}}{5} \\ \tan \theta &= 2 \\ \cot \theta &= \frac{1}{2} \\ \sec \theta &= \sqrt{5} \\ \csc \theta &= \frac{\sqrt{5}}{2} \\ \end{aligned} \]