WEBVTT

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Here is the problem: sin(x/2) equals 1 over 2.

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A periodic equation in x: it has infinitely many solutions, one family for each solution in a single period.

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Start with the equation.

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Rewrite as f(x) equals 0.

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The left side repeats every 4pi, so find the solutions in one period. f(x plus 4pi) equals f(x) for every x, so every solution is one of the solutions in [0, 4pi) plus a whole number of periods.

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Solutions in one period [0, 4pi).

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Add any whole number of periods.

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So the answer is x equals pi/3 plus 4pik, x equals 5pi/3 plus 4pik (k any integer).

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Each solution in one period was found numerically, matched to an exact value and confirmed by substitution; adding whole periods gives all solutions.

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x equals 1.0471976: left side equals 0.5, right side equals 0.5, which checks out.

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x equals 5.2359878: left side equals 0.5, right side equals 0.5, which checks out.