1. The problem is to find the values of cosec, sec, and cot for the angles $0^\circ$, $30^\circ$, $45^\circ$, $60^\circ$, and $90^\circ$. 2. Recall the definitions: - $\csc \theta = \frac{1}{\sin \theta}$ - $\sec \theta = \frac{1}{\cos \theta}$ - $\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}$ 3. Use known sine, cosine, and tangent values for these angles: - $\sin 0^\circ = 0$, $\cos 0^\circ = 1$, $\tan 0^\circ = 0$ - $\sin 30^\circ = \frac{1}{2}$, $\cos 30^\circ = \frac{\sqrt{3}}{2}$, $\tan 30^\circ = \frac{1}{\sqrt{3}}$ - $\sin 45^\circ = \frac{\sqrt{2}}{2}$, $\cos 45^\circ = \frac{\sqrt{2}}{2}$, $\tan 45^\circ = 1$ - $\sin 60^\circ = \frac{\sqrt{3}}{2}$, $\cos 60^\circ = \frac{1}{2}$, $\tan 60^\circ = \sqrt{3}$ - $\sin 90^\circ = 1$, $\cos 90^\circ = 0$, $\tan 90^\circ$ is undefined. 4. Calculate each: - For $0^\circ$: - $\csc 0^\circ = \frac{1}{0}$ undefined - $\sec 0^\circ = \frac{1}{1} = 1$ - $\cot 0^\circ = \frac{1}{0}$ undefined - For $30^\circ$: - $\csc 30^\circ = \frac{1}{\frac{1}{2}} = 2$ - $\sec 30^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$ after rationalizing - $\cot 30^\circ = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3}$ - For $45^\circ$: - $\csc 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$ after rationalizing - $\sec 45^\circ = \frac{1}{\frac{\sqrt{2}}{2}} = \sqrt{2}$ - $\cot 45^\circ = \frac{1}{1} = 1$ - For $60^\circ$: - $\csc 60^\circ = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$ after rationalizing - $\sec 60^\circ = \frac{1}{\frac{1}{2}} = 2$ - $\cot 60^\circ = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ after rationalizing - For $90^\circ$: - $\csc 90^\circ = \frac{1}{1} = 1$ - $\sec 90^\circ = \frac{1}{0}$ undefined - $\cot 90^\circ = \frac{1}{\tan 90^\circ}$ undefined 5. Summary table: | Angle | $\csc$ | $\sec$ | $\cot$ | |-------|--------|--------|--------| | $0^\circ$ | undefined | 1 | undefined | | $30^\circ$ | 2 | $\frac{2\sqrt{3}}{3}$ | $\sqrt{3}$ | | $45^\circ$ | $\sqrt{2}$ | $\sqrt{2}$ | 1 | | $60^\circ$ | $\frac{2\sqrt{3}}{3}$ | 2 | $\frac{\sqrt{3}}{3}$ | | $90^\circ$ | 1 | undefined | undefined |