WEBVTT

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Here is the problem: Find arctan(negative 1) and every angle in [0°, 360°) with tan x equals negative 1.

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Find the angle whose tan is negative 1: the principal value of arctan, every solution of tan(x) equals negative 1 in one turn, and the general solution.

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Principal value of arctan, taken in the range (−pi/2, pi/2). negative 45° is a special angle, so the answer is exact.

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tan repeats every 180°, so add multiples of 180°.

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Solutions in one full turn, 0° less than or equal to x less than 360°.

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General solution, n any integer.

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So the answer is arctan(negative 1) equals negative 45° (-pi/4); solutions in [0°, 360°): 135°, 315°; general: x equals negative 45° plus 180°n.

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An inverse trigonometric function returns one angle, the principal value. Because the trigonometric functions repeat, the equation tan(x) equals negative 1 has infinitely many solutions; the calculator lists those in one turn and the general pattern.

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Substituting each solution back: tan(135°) approximately negative 1; tan(315°) approximately negative 1.