1. **State the six trigonometric functions for point (-5, -12):** - Step 1: Calculate the radius $r = \sqrt{x^2 + y^2} = \sqrt{(-5)^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13$. - Step 2: Define the functions: - $\sin \theta = \frac{y}{r} = \frac{-12}{13}$ - $\cos \theta = \frac{x}{r} = \frac{-5}{13}$ - $\tan \theta = \frac{y}{x} = \frac{-12}{-5} = \frac{12}{5}$ - $\csc \theta = \frac{r}{y} = \frac{13}{-12} = -\frac{13}{12}$ - $\sec \theta = \frac{r}{x} = \frac{13}{-5} = -\frac{13}{5}$ - $\cot \theta = \frac{x}{y} = \frac{-5}{-12} = \frac{5}{12}$ 2. **State the six trigonometric functions for point (-1, 0):** - Step 1: Calculate $r = \sqrt{(-1)^2 + 0^2} = 1$. - Step 2: Define the functions: - $\sin \theta = \frac{0}{1} = 0$ - $\cos \theta = \frac{-1}{1} = -1$ - $\tan \theta = \frac{0}{-1} = 0$ - $\csc \theta = \frac{1}{0}$ (undefined) - $\sec \theta = \frac{1}{-1} = -1$ - $\cot \theta = \frac{-1}{0}$ (undefined) 3. **Find exact values for $\theta = \frac{7\pi}{6}$ using reference angles:** - Step 1: Reference angle is $\pi - \frac{7\pi}{6} = \frac{\pi}{6}$. - Step 2: Since $\frac{7\pi}{6}$ is in the third quadrant, sine and cosine are negative. - Step 3: Use known values: - $\sin \frac{7\pi}{6} = -\sin \frac{\pi}{6} = -\frac{1}{2}$ - $\cos \frac{7\pi}{6} = -\cos \frac{\pi}{6} = -\frac{\sqrt{3}}{2}$ - $\tan \frac{7\pi}{6} = \frac{\sin}{\cos} = \frac{-\frac{1}{2}}{-\frac{\sqrt{3}}{2}} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ 4. **Find exact values for $\theta = -\frac{3\pi}{4}$ using reference angles:** - Step 1: Add $2\pi$ to get positive angle: $-\frac{3\pi}{4} + 2\pi = \frac{5\pi}{4}$. - Step 2: Reference angle is $\frac{5\pi}{4} - \pi = \frac{\pi}{4}$. - Step 3: $\frac{5\pi}{4}$ is in the third quadrant, sine and cosine are negative. - Step 4: Use known values: - $\sin -\frac{3\pi}{4} = \sin \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$ - $\cos -\frac{3\pi}{4} = \cos \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$ - $\tan -\frac{3\pi}{4} = \tan \frac{5\pi}{4} = 1$ 5. **Use calculator in radian mode:** - Convert $121^\circ$ to radians: $121^\circ \times \frac{\pi}{180} = \frac{121\pi}{180}$ (approx 2.11 radians). - Evaluate: - $\sin 121^\circ \approx \sin 2.11 \approx 0.8572$ - $\cos -\frac{3\pi}{5} = \cos -1.884 \approx -0.3090$ - $\cot 3 = \frac{1}{\tan 3} \approx \frac{1}{0.1425} \approx 7.015$ 6. **Given $\sin \theta = \frac{1}{3}$ and $\cos \theta < 0$, find $\tan \theta$ and $\sec \theta$:** - Step 1: Use Pythagorean identity: $\sin^2 \theta + \cos^2 \theta = 1$. - Step 2: Calculate $\cos \theta = -\sqrt{1 - \sin^2 \theta} = -\sqrt{1 - \left(\frac{1}{3}\right)^2} = -\sqrt{1 - \frac{1}{9}} = -\sqrt{\frac{8}{9}} = -\frac{2\sqrt{2}}{3}$. - Step 3: Calculate $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{1}{3}}{-\frac{2\sqrt{2}}{3}} = -\frac{1}{2\sqrt{2}} = -\frac{\sqrt{2}}{4}$ after rationalizing. - Step 4: Calculate $\sec \theta = \frac{1}{\cos \theta} = -\frac{3}{2\sqrt{2}} = -\frac{3\sqrt{2}}{4}$ after rationalizing. 7. **Use unit circle to find $\csc 60^\circ$:** - Step 1: $\sin 60^\circ = \frac{\sqrt{3}}{2}$. - Step 2: $\csc 60^\circ = \frac{1}{\sin 60^\circ} = \frac{1}{\frac{\sqrt{3}}{2}} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$ after rationalizing. 8. **Find $\cot \frac{\pi}{6}$:** - Step 1: $\tan \frac{\pi}{6} = \frac{1}{\sqrt{3}}$. - Step 2: $\cot \frac{\pi}{6} = \frac{1}{\tan \frac{\pi}{6}} = \sqrt{3}$. 9. **Find $\cos \left(-\frac{23\pi}{6}\right)$:** - Step 1: Add $4\pi$ (or $\frac{24\pi}{6}$) to get positive angle: $-\frac{23\pi}{6} + \frac{24\pi}{6} = \frac{\pi}{6}$. - Step 2: $\cos \left(-\frac{23\pi}{6}\right) = \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}$. 10. **Find $\sin \left(-\frac{19\pi}{4}\right)$:** - Step 1: Add $6\pi$ (or $\frac{24\pi}{4}$) to get positive angle: $-\frac{19\pi}{4} + \frac{24\pi}{4} = \frac{5\pi}{4}$. - Step 2: $\sin \left(-\frac{19\pi}{4}\right) = \sin \frac{5\pi}{4} = -\frac{\sqrt{2}}{2}$. This completes the study guide with detailed explanations and calculations for each problem.