1. **Problem Statement:** Given a unit circle centered at the origin, the point (1, 0) is rotated by an angle $\theta$ about the origin. We need to find: a) For which values of $\theta$ the y-coordinate of the rotated point is positive or negative. b) The relationship between the y-coordinate and $\sin(\theta)$. c) Compute $\sin(90^\circ)$, $\sin(180^\circ)$, and $\sin(270^\circ)$ using this relationship. 2. **Formula and Important Rules:** The coordinates of a point on the unit circle after rotation by angle $\theta$ are given by: $$ (x, y) = (\cos(\theta), \sin(\theta)) $$ The y-coordinate of the rotated point is therefore $y = \sin(\theta)$. 3. **Part a) Values of $\theta$ for positive and negative y-coordinate:** - The y-coordinate is positive when $\sin(\theta) > 0$. - The y-coordinate is negative when $\sin(\theta) < 0$. On the unit circle: - $\sin(\theta) > 0$ for angles in the first and second quadrants, i.e., $$ 0^\circ < \theta < 180^\circ $$ - $\sin(\theta) < 0$ for angles in the third and fourth quadrants, i.e., $$ 180^\circ < \theta < 360^\circ $$ 4. **Part b) Relationship between y-coordinate and $\sin(\theta)$:** The y-coordinate of the rotated point on the unit circle is exactly equal to $\sin(\theta)$. This means: $$ y = \sin(\theta) $$ This relationship holds for all angles $\theta$. 5. **Part c) Compute $\sin(90^\circ)$, $\sin(180^\circ)$, and $\sin(270^\circ)$:** Using the unit circle and the relationship $y = \sin(\theta)$: - At $90^\circ$, the point is at $(0,1)$, so: $$ \sin(90^\circ) = 1 $$ - At $180^\circ$, the point is at $(-1,0)$, so: $$ \sin(180^\circ) = 0 $$ - At $270^\circ$, the point is at $(0,-1)$, so: $$ \sin(270^\circ) = -1 $$ **Final answers:** - a) $y > 0$ for $0^\circ < \theta < 180^\circ$, $y < 0$ for $180^\circ < \theta < 360^\circ$. - b) $y = \sin(\theta)$. - c) $\sin(90^\circ) = 1$, $\sin(180^\circ) = 0$, $\sin(270^\circ) = -1$.