1. **Problem Statement:** We have 6 LEDs placed evenly around a circle of radius 100 inches centered at the origin (0,0). The LEDs are at angles 0°, 60°, 120°, 180°, 240°, and 300°. We need to find their coordinates using vector representation and trigonometric functions. 2. **Formula and Explanation:** Each LED position can be represented as a vector from the origin with length (magnitude) 100 inches and angle $\theta$. The coordinates $(x,y)$ of a point on a circle with radius $r$ and angle $\theta$ are given by: $$ x = r \cos(\theta) \\ y = r \sin(\theta) $$ where $\theta$ is in degrees (converted to radians for calculation). 3. **Calculate Coordinates:** - LED A at $0^\circ$: $$x = 100 \cos(0^\circ) = 100 \times 1 = 100$$ $$y = 100 \sin(0^\circ) = 100 \times 0 = 0$$ Coordinates: $(100, 0)$ - LED B at $60^\circ$: $$x = 100 \cos(60^\circ) = 100 \times 0.5 = 50$$ $$y = 100 \sin(60^\circ) = 100 \times \frac{\sqrt{3}}{2} \approx 86.6$$ Coordinates: $(50, 87)$ (rounded) - LED C at $120^\circ$: $$x = 100 \cos(120^\circ) = 100 \times (-0.5) = -50$$ $$y = 100 \sin(120^\circ) = 100 \times \frac{\sqrt{3}}{2} \approx 86.6$$ Coordinates: $(-50, 87)$ - LED D at $180^\circ$: $$x = 100 \cos(180^\circ) = 100 \times (-1) = -100$$ $$y = 100 \sin(180^\circ) = 100 \times 0 = 0$$ Coordinates: $(-100, 0)$ - LED E at $240^\circ$: $$x = 100 \cos(240^\circ) = 100 \times (-0.5) = -50$$ $$y = 100 \sin(240^\circ) = 100 \times (-\frac{\sqrt{3}}{2}) \approx -86.6$$ Coordinates: $(-50, -87)$ - LED F at $300^\circ$: $$x = 100 \cos(300^\circ) = 100 \times 0.5 = 50$$ $$y = 100 \sin(300^\circ) = 100 \times (-\frac{\sqrt{3}}{2}) \approx -86.6$$ Coordinates: $(50, -87)$ 4. **Summary:** The LED coordinates rounded to integers are: - A: $(100, 0)$ - B: $(50, 87)$ - C: $(-50, 87)$ - D: $(-100, 0)$ - E: $(-50, -87)$ - F: $(50, -87)$ These coordinates describe the LED positions on the 200" diagonal image in inches. 5. **Additional Notes:** - Using coordinate systems allows precise and consistent description of LED positions. - The screen coordinate system is typically Cartesian with origin at top-left, but here the origin is at the center. - Screen information is stored as pixel coordinates. - The vector length (100") and angle steps (60°) relate to sine and cosine functions to find exact positions. Final answer: $$ \boxed{\begin{cases} A = (100, 0) \\ B = (50, 87) \\ C = (-50, 87) \\ D = (-100, 0) \\ E = (-50, -87) \\ F = (50, -87) \end{cases}} $$