WEBVTT

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Here is the problem: tan(3x) equals sqrt(3).

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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 pi/3, so find the solutions in one period. f(x plus pi/3) equals f(x) for every x, so every solution is one of the solutions in [0, pi/3) plus a whole number of periods.

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Solutions in one period [0, pi/3).

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Add any whole number of periods.

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So the answer is x equals pi/9 plus pik/3 (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 0.34906585: left side equals 1.7320508, right side equals 1.7320508, which checks out.