1. **Convert 576° to radians, leaving your answer in terms of π.** The formula to convert degrees to radians is: $$\text{radians} = \text{degrees} \times \frac{\pi}{180}$$ So, $$576^\circ \times \frac{\pi}{180} = \frac{576\pi}{180}$$ Simplify the fraction by dividing numerator and denominator by 12: $$\frac{\cancel{576}^{48}\pi}{\cancel{180}^{15}} = \frac{48\pi}{15}$$ Further simplify by dividing numerator and denominator by 3: $$\frac{\cancel{48}^{16}\pi}{\cancel{15}^{5}} = \frac{16\pi}{5}$$ **Answer:** $$576^\circ = \frac{16\pi}{5} \text{ radians}$$ 2. **Express $\frac{2\pi}{9}$ radians in degrees.** The formula to convert radians to degrees is: $$\text{degrees} = \text{radians} \times \frac{180}{\pi}$$ So, $$\frac{2\pi}{9} \times \frac{180}{\pi} = \frac{2\cancel{\pi}}{9} \times \frac{180}{\cancel{\pi}} = \frac{2 \times 180}{9} = \frac{360}{9} = 40^\circ$$ **Answer:** $$\frac{2\pi}{9} \text{ radians} = 40^\circ$$ 3. **An airplane is flying 3750 m above ground. The pilot observes an airport at a distance of 4635 m.** Given a right triangle with vertical side (height) = 3750 m and hypotenuse = 4635 m. (a) Calculate the angle of depression of the airport from the plane. The angle of depression corresponds to the angle between the hypotenuse and the horizontal line from the plane. Use cosine: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\text{horizontal distance}}{4635}$$ But we don't know horizontal distance yet. Alternatively, use sine with opposite side (height): $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3750}{4635}$$ Calculate: $$\sin(\theta) = \frac{3750}{4635} \approx 0.8089$$ Find angle: $$\theta = \sin^{-1}(0.8089) \approx 54.0^\circ$$ **Answer:** Angle of depression $\theta \approx 54.0^\circ$ (b) Calculate the horizontal distance between the plane and the airport. Use Pythagoras theorem or cosine: $$\cos(\theta) = \frac{\text{horizontal distance}}{4635}$$ Calculate cosine: $$\cos(54.0^\circ) \approx 0.5878$$ So, $$\text{horizontal distance} = 4635 \times 0.5878 \approx 2723.5 \text{ m}$$ **Answer:** Horizontal distance $\approx 2723.5$ meters