1. **Express each angle in radian measure** The formula to convert degrees to radians is: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$ We multiply each degree value by $\frac{\pi}{180}$ to get radians. - a. $60^\circ = 60 \times \frac{\pi}{180} = \frac{\cancel{60}}{\cancel{180}}\pi = \frac{\pi}{3}$ - b. $-72^\circ = -72 \times \frac{\pi}{180} = -\frac{\cancel{72}}{\cancel{180}}\pi = -\frac{2\pi}{5}$ - c. $135^\circ = 135 \times \frac{\pi}{180} = \frac{\cancel{135}}{\cancel{180}}\pi = \frac{3\pi}{4}$ - d. $36^\circ = 36 \times \frac{\pi}{180} = \frac{\cancel{36}}{\cancel{180}}\pi = \frac{\pi}{5}$ - e. $-80^\circ = -80 \times \frac{\pi}{180} = -\frac{4\pi}{9}$ - f. $200^\circ = 200 \times \frac{\pi}{180} = \frac{10\pi}{9}$ - g. $155^\circ = 155 \times \frac{\pi}{180} = \frac{31\pi}{36}$ - h. $-150^\circ = -150 \times \frac{\pi}{180} = -\frac{5\pi}{6}$ - i. $40^\circ = 40 \times \frac{\pi}{180} = \frac{2\pi}{9}$ - j. $-225^\circ = -225 \times \frac{\pi}{180} = -\frac{5\pi}{4}$ 2. **Express each angle in degrees** The formula to convert radians to degrees is: $$\text{degrees} = \text{radians} \times \frac{180}{\pi}$$ Multiply each radian value by $\frac{180}{\pi}$. - a. $\frac{3\pi}{4} = \frac{3\pi}{4} \times \frac{180}{\pi} = 135^\circ$ - b. $\frac{5\pi}{3} = \frac{5\pi}{3} \times \frac{180}{\pi} = 300^\circ$ - c. $5\pi = 5\pi \times \frac{180}{\pi} = 900^\circ$ - d. $-\frac{\pi}{12} = -\frac{\pi}{12} \times \frac{180}{\pi} = -15^\circ$ - e. $\frac{17\pi}{6} = \frac{17\pi}{6} \times \frac{180}{\pi} = 510^\circ$ - f. $\frac{5\pi}{6} = \frac{5\pi}{6} \times \frac{180}{\pi} = 150^\circ$ - g. $-\frac{7\pi}{2} = -\frac{7\pi}{2} \times \frac{180}{\pi} = -630^\circ$ - h. $\frac{2\pi}{9} = \frac{2\pi}{9} \times \frac{180}{\pi} = 40^\circ$ - i. $\frac{\pi}{18} = \frac{\pi}{18} \times \frac{180}{\pi} = 10^\circ$ - j. $-\frac{9\pi}{5} = -\frac{9\pi}{5} \times \frac{180}{\pi} = -324^\circ$ 3. **Find positive and negative coterminal angles** Coterminal angles differ by full rotations: $360^\circ$ or $2\pi$ radians. - a. $733^\circ$ - Positive coterminal: $733 - 360 = 373^\circ$ - Negative coterminal: $733 - 2 \times 360 = 13^\circ$ - b. $-100^\circ$ - Positive coterminal: $-100 + 360 = 260^\circ$ - Negative coterminal: $-100 - 360 = -460^\circ$ - c. $\frac{5\pi}{3}$ - Positive coterminal: $\frac{5\pi}{3} + 2\pi = \frac{5\pi}{3} + \frac{6\pi}{3} = \frac{11\pi}{3}$ - Negative coterminal: $\frac{5\pi}{3} - 2\pi = \frac{5\pi}{3} - \frac{6\pi}{3} = -\frac{\pi}{3}$ - d. $-\frac{2\pi}{3}$ - Positive coterminal: $-\frac{2\pi}{3} + 2\pi = \frac{4\pi}{3}$ - Negative coterminal: $-\frac{2\pi}{3} - 2\pi = -\frac{8\pi}{3}$ 4. **Find exact values of trigonometric functions** Use unit circle values: - a. $\sin(-\frac{\pi}{3}) = -\sin(\frac{\pi}{3}) = -\frac{\sqrt{3}}{2}$ - b. $\tan(\frac{\pi}{4}) = 1$ - c. $\sec(\pi) = \frac{1}{\cos(\pi)} = \frac{1}{-1} = -1$ - d. $\sin(-\frac{3\pi}{2}) = \sin(\frac{\pi}{2}) = 1$ - e. $\cos(-315^\circ) = \cos(45^\circ) = \frac{\sqrt{2}}{2}$ - f. $\cos(\frac{9\pi}{4}) = \cos(2\pi + \frac{\pi}{4}) = \cos(\frac{\pi}{4}) = \frac{\sqrt{2}}{2}$ - g. $\cot(-\frac{\pi}{3}) = \cot(\frac{\pi}{3}) = \frac{1}{\tan(\frac{\pi}{3})} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$ - h. $\csc(\frac{5\pi}{6}) = \frac{1}{\sin(\frac{5\pi}{6})} = \frac{1}{\frac{1}{2}} = 2$ - i. $\tan(135^\circ) = \tan(\pi - \frac{\pi}{4}) = -\tan(\frac{\pi}{4}) = -1$ - j. $\cot(330^\circ) = \cot(2\pi - \frac{\pi}{6}) = -\cot(\frac{\pi}{6}) = -\sqrt{3}$ 5. **Find amplitude, period, and phase shift** General form: $y = A \sin(Bx - C)$ or $y = A \cos(Bx + C)$ - Amplitude = $|A|$ - Period = $\frac{2\pi}{|B|}$ - Phase shift = $\frac{C}{B}$ (sign changes if inside parentheses is $x - C/B$ or $x + C/B$) - a. $y = \frac{2}{3} \sin(2x - \frac{3\pi}{4})$ - Amplitude = $\frac{2}{3}$ - Period = $\frac{2\pi}{2} = \pi$ - Phase shift = $\frac{3\pi/4}{2} = \frac{3\pi}{8}$ to the right - b. $y = -3 \cos(\frac{1}{2}x + \pi)$ - Amplitude = $|-3| = 3$ - Period = $\frac{2\pi}{\frac{1}{2}} = 4\pi$ - Phase shift = $-\frac{\pi}{\frac{1}{2}} = -2\pi$ (shift left) Final answers are summarized in the steps above.