1. **State the problem:** We need to find the values of the cotangent function for the angles 0°, 30°, 45°, 60°, and 90°. 2. **Recall the definition:** The cotangent of an angle $\theta$ is defined as $$\cot \theta = \frac{\cos \theta}{\sin \theta}$$ 3. **Evaluate each angle:** - $\cot 0^\circ = \frac{\cos 0^\circ}{\sin 0^\circ} = \frac{1}{0}$ which is **not defined** because division by zero is undefined. - $\cot 30^\circ = \frac{\cos 30^\circ}{\sin 30^\circ} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3}$ - $\cot 45^\circ = \frac{\cos 45^\circ}{\sin 45^\circ} = \frac{\frac{\sqrt{2}}{2}}{\frac{\sqrt{2}}{2}} = 1$ - $\cot 60^\circ = \frac{\cos 60^\circ}{\sin 60^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ (rationalized) - $\cot 90^\circ = \frac{\cos 90^\circ}{\sin 90^\circ} = \frac{0}{1} = 0$ 4. **Summary:** - $\cot 0^\circ$ is not defined - $\cot 30^\circ = \sqrt{3}$ - $\cot 45^\circ = 1$ - $\cot 60^\circ = \frac{\sqrt{3}}{3}$ - $\cot 90^\circ = 0$ These values match the second group given in the problem. **Final answer:** $$\cot 30^\circ = \sqrt{3}, \quad \cot 60^\circ = \frac{\sqrt{3}}{3}, \quad \cot 90^\circ = 0, \quad \cot 45^\circ = 1, \quad \cot 0^\circ \text{ is not defined}$$