1. **Problem Statement:** Given angles in degrees, we need to: - Sketch each angle in standard position. - Find the exact values of the six trigonometric functions: $\sin$, $\cos$, $\tan$, $\csc$, $\sec$, and $\cot$. - Convert each angle to radian measure. 2. **Formula and Rules:** - To convert degrees to radians: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$ - The six trig functions are defined as: $$\sin \theta = \frac{y}{r}, \quad \cos \theta = \frac{x}{r}, \quad \tan \theta = \frac{y}{x}$$ $$\csc \theta = \frac{1}{\sin \theta}, \quad \sec \theta = \frac{1}{\cos \theta}, \quad \cot \theta = \frac{1}{\tan \theta}$$ - Angles can be reduced to their reference angle in the first rotation by subtracting or adding multiples of 360°. --- ### a. Angle: 390° 1. Convert to standard position by subtracting 360°: $$390^\circ - 360^\circ = 30^\circ$$ 2. Convert to radians: $$390^\circ \times \frac{\pi}{180} = \frac{390\pi}{180} = \frac{13\pi}{6}$$ 3. Reference angle is $30^\circ$. 4. Trig values for $30^\circ$: $$\sin 30^\circ = \frac{1}{2}, \quad \cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \tan 30^\circ = \frac{1}{\sqrt{3}}$$ 5. Since 390° is in Quadrant I, signs are positive. 6. Reciprocal functions: $$\csc 390^\circ = 2, \quad \sec 390^\circ = \frac{2}{\sqrt{3}}, \quad \cot 390^\circ = \sqrt{3}$$ --- ### b. Angle: 510° 1. Reduce by subtracting 360°: $$510^\circ - 360^\circ = 150^\circ$$ 2. Convert to radians: $$510^\circ \times \frac{\pi}{180} = \frac{17\pi}{6}$$ 3. Reference angle is $180^\circ - 150^\circ = 30^\circ$. 4. Trig values for $150^\circ$ (Quadrant II): $$\sin 150^\circ = \frac{1}{2}, \quad \cos 150^\circ = -\frac{\sqrt{3}}{2}, \quad \tan 150^\circ = -\frac{1}{\sqrt{3}}$$ 5. Reciprocal functions: $$\csc 150^\circ = 2, \quad \sec 150^\circ = -\frac{2}{\sqrt{3}}, \quad \cot 150^\circ = -\sqrt{3}$$ --- ### c. Angle: 240° 1. Convert to radians: $$240^\circ \times \frac{\pi}{180} = \frac{4\pi}{3}$$ 2. Reference angle: $$240^\circ - 180^\circ = 60^\circ$$ 3. Trig values for $240^\circ$ (Quadrant III): $$\sin 240^\circ = -\frac{\sqrt{3}}{2}, \quad \cos 240^\circ = -\frac{1}{2}, \quad \tan 240^\circ = \sqrt{3}$$ 4. Reciprocal functions: $$\csc 240^\circ = -\frac{2}{\sqrt{3}}, \quad \sec 240^\circ = -2, \quad \cot 240^\circ = \frac{1}{\sqrt{3}}$$ --- ### d. Angle: -690° 1. Add 360° twice to get positive coterminal angle: $$-690^\circ + 720^\circ = 30^\circ$$ 2. Convert to radians: $$-690^\circ \times \frac{\pi}{180} = -\frac{23\pi}{6}$$ 3. Trig values same as $30^\circ$ (Quadrant I): $$\sin = \frac{1}{2}, \quad \cos = \frac{\sqrt{3}}{2}, \quad \tan = \frac{1}{\sqrt{3}}$$ 4. Reciprocal functions: $$\csc = 2, \quad \sec = \frac{2}{\sqrt{3}}, \quad \cot = \sqrt{3}$$ --- ### e. Angle: -150° 1. Add 360° to get positive coterminal angle: $$-150^\circ + 360^\circ = 210^\circ$$ 2. Convert to radians: $$-150^\circ \times \frac{\pi}{180} = -\frac{5\pi}{6}$$ 3. Reference angle: $$210^\circ - 180^\circ = 30^\circ$$ 4. Trig values for $210^\circ$ (Quadrant III): $$\sin 210^\circ = -\frac{1}{2}, \quad \cos 210^\circ = -\frac{\sqrt{3}}{2}, \quad \tan 210^\circ = \frac{1}{\sqrt{3}}$$ 5. Reciprocal functions: $$\csc 210^\circ = -2, \quad \sec 210^\circ = -\frac{2}{\sqrt{3}}, \quad \cot 210^\circ = \sqrt{3}$$