Solving sin(x) = 1/2

Trigonometry · Axiom Academy

Finding all angles in [0, 2π) where sine equals 1/2 using the unit circle and reference angles 1. The Equation and What It Means The sine of an angle is the y-coordinate of the corresponding point on the unit circle. So we need to find where the unit circle intersects the horizontal line y = 1/2. 2. Finding the First Solution (Quadrant I) The first intersection point occurs in Quadrant I , where both x and y coordinates are positive. From our knowledge of special angles: This is one of the fundamental angles you should memorize. The angle π/6 (or 30°) creates a point on the unit circle with y-coordinate of exactly 1/2. 3. Finding the Second Solution (Quadrant II) But wait! The horizontal line y = 1/2 intersects the circle at two points . The second intersection is in Quadrant II, where x is negative but y is still positive. We've found both angles in the interval [0, 2π) where sin(x) = 1/2. Let's verify and summarize our solutions:

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