1. Find two positive coterminal angles of $-135^\circ$. Coterminal angles differ by multiples of $360^\circ$. Step 1: Add $360^\circ$ to $-135^\circ$: $$-135^\circ + 360^\circ = 225^\circ$$ Step 2: Add another $360^\circ$ for the second positive coterminal angle: $$225^\circ + 360^\circ = 585^\circ$$ Answer: Two positive coterminal angles are $225^\circ$ and $585^\circ$. 2. Find one negative coterminal angle of $480^\circ$. Step 1: Subtract $360^\circ$ to get a coterminal angle: $$480^\circ - 360^\circ = 120^\circ$$ (positive) Step 2: Subtract another $360^\circ$ for a negative coterminal angle: $$120^\circ - 360^\circ = -240^\circ$$ Answer: One negative coterminal angle is $-240^\circ$. 3. Convert $210^\circ$ to radians in simplest fractional form. Step 1: Use conversion: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$ Step 2: Apply conversion: $$210^\circ \times \frac{\pi}{180} = \frac{210\pi}{180} = \frac{7\pi}{6}$$ Answer: $210^\circ = \frac{7\pi}{6}$ radians. 4. Convert $\frac{7\pi}{4}$ radians to degrees. Step 1: Use conversion: $$\text{degrees} = \text{radians} \times \frac{180}{\pi}$$ Step 2: Apply conversion: $$\frac{7\pi}{4} \times \frac{180}{\pi} = \frac{7 \times 180}{4} = 315^\circ$$ Answer: $\frac{7\pi}{4}$ radians = $315^\circ$. 5. Convert $45^\circ$ and $135^\circ$ to radians, simplify answers. For $45^\circ$: $$45^\circ \times \frac{\pi}{180} = \frac{45\pi}{180} = \frac{\pi}{4}$$ For $135^\circ$: $$135^\circ \times \frac{\pi}{180} = \frac{135\pi}{180} = \frac{3\pi}{4}$$ Answer: $45^\circ = \frac{\pi}{4}$ radians and $135^\circ = \frac{3\pi}{4}$ radians. 6. Convert $\frac{2\pi}{3}$ and $\frac{11\pi}{6}$ radians to degrees. For $\frac{2\pi}{3}$: $$\frac{2\pi}{3} \times \frac{180}{\pi} = \frac{2 \times 180}{3} = 120^\circ$$ For $\frac{11\pi}{6}$: $$\frac{11\pi}{6} \times \frac{180}{\pi} = \frac{11 \times 180}{6} = 330^\circ$$ Answer: $\frac{2\pi}{3} = 120^\circ$ and $\frac{11\pi}{6} = 330^\circ$. 7. In a right triangle, opposite side $= 8$, hypotenuse $= 17$, find $\sin \theta$ and $\csc \theta$. Step 1: Recall $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{8}{17}$. Step 2: $\csc \theta$ is the reciprocal: $$\csc \theta = \frac{1}{\sin \theta} = \frac{17}{8}$$ Answer: $\sin \theta = \frac{8}{17}$ and $\csc \theta = \frac{17}{8}$. 8. Given $\cos \theta = \frac{3}{5}$, calculate $\sec \theta$ and $\tan \theta$. Step 1: $\sec \theta = \frac{1}{\cos \theta} = \frac{1}{3/5} = \frac{5}{3}$. Step 2: Use identity $\sin^2 \theta + \cos^2 \theta = 1$ $$\sin \theta = \sqrt{1 - \cos^2 \theta} = \sqrt{1 - \left(\frac{3}{5}\right)^2} = \sqrt{1 - \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5}$$ Step 3: Compute $\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{4/5}{3/5} = \frac{4}{3}$. Answer: $\sec \theta = \frac{5}{3}$ and $\tan \theta = \frac{4}{3}$. 9. Given $\tan \theta = 3$ and $\cos \theta = \frac{1}{\sqrt{10}}$, find $\sin \theta$ and $\cot \theta$. Step 1: Recall $\cot \theta = \frac{1}{\tan \theta} = \frac{1}{3}$. Step 2: Use identity $\tan \theta = \frac{\sin \theta}{\cos \theta}$ to find $\sin \theta$: $$\sin \theta = \tan \theta \times \cos \theta = 3 \times \frac{1}{\sqrt{10}} = \frac{3}{\sqrt{10}}$$ Answer: $\sin \theta = \frac{3}{\sqrt{10}}$ and $\cot \theta = \frac{1}{3}$. 10. Given $\tan \theta = \frac{5}{12}$ and $\cos \theta = \frac{12}{13}$, find $\sin \theta$ and $\cot \theta$. Step 1: $\cot \theta = \frac{1}{\tan \theta} = \frac{12}{5}$. Step 2: Use $\tan \theta = \frac{\sin \theta}{\cos \theta}$: $$\sin \theta = \tan \theta \times \cos \theta = \frac{5}{12} \times \frac{12}{13} = \frac{5}{13}$$ Answer: $\sin \theta = \frac{5}{13}$ and $\cot \theta = \frac{12}{5}$.