1. **Problem Statement:** Given a circle centered at the origin and a point on the circle at coordinates $(10, 24)$, find: a) $\sin(\theta)$ where $\theta$ is the angle between the positive x-axis and the radius to the point. b) The measure of angle $w$ in degrees. c) The measure of angle $\theta$ in degrees. 2. **Formula and Important Rules:** - The radius $r$ of the circle is the distance from the origin to the point $(10, 24)$, calculated by the distance formula: $$r = \sqrt{x^2 + y^2}$$ - The sine of angle $\theta$ in a right triangle is the ratio of the opposite side to the hypotenuse: $$\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$$ - The angle $w$ is complementary to $\theta$ in the right triangle, so: $$m\angle w = 90^\circ - m\angle \theta$$ 3. **Calculate the radius $r$:** $$r = \sqrt{10^2 + 24^2} = \sqrt{100 + 576} = \sqrt{676} = 26$$ 4. **Calculate $\sin(\theta)$:** The opposite side to $\theta$ is the y-coordinate 24, and the hypotenuse is $r=26$: $$\sin(\theta) = \frac{24}{26} = \frac{\cancel{24}}{\cancel{26}} = \frac{12}{13}$$ 5. **Calculate $m\angle \theta$ in degrees:** Use the inverse sine function: $$m\angle \theta = \sin^{-1}\left(\frac{12}{13}\right) \approx 67.38^\circ$$ 6. **Calculate $m\angle w$ in degrees:** Since $w$ and $\theta$ are complementary: $$m\angle w = 90^\circ - 67.38^\circ = 22.62^\circ$$ **Final answers:** - a) $\sin(\theta) = \frac{12}{13}$ - b) $m\angle w = 22.6$ deg (rounded to one decimal place) - c) $m\angle \theta = 67.4$ deg (rounded to one decimal place)