WEBVTT

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Here is the problem: tan(2x) equals 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 pi/2, so find the solutions in one period. f(x plus pi/2) equals f(x) for every x, so every solution is one of the solutions in [0, pi/2) plus a whole number of periods.

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Solutions in one period [0, pi/2).

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Add any whole number of periods.

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So the answer is x approximately 0.5535743589 plus pik/2 (k any integer).

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Each solution in one period is a numerical approximation (about 10 significant digits) confirmed by substitution; adding whole periods gives all solutions.

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x equals 0.55357436: left side equals 2, right side equals 2, which checks out.