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Geometry and Trigonometry / Right triangles and trigonometry Difficulty: Hard

In a right triangle, the tangent of one of the two acute angles is $the fraction with numerator the square root of 3, and denominator 3$ . What is the tangent of the other acute angle?

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Explanation

Choice D is correct. The tangent of a nonright angle in a right triangle is defined as the ratio of the length of the leg opposite the angle to the length of the leg adjacent to the angle. Using that definition for tangent, in a right triangle with legs that have lengths a and b, the tangent of one acute angle is $the fraction a, over b$ and the tangent for the other acute angle is $the fraction b over a$ . It follows that the tangents of the acute angles in a right triangle are reciprocals of each other. Therefore, the tangent of the other acute angle in the given triangle is the reciprocal of $the fraction, the square root of 3, end root, over 3, end fraction$ or $the fraction, 3 over the square root of 3, end fraction$ .

Choice A is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the tangent of the angle described. Choice B is incorrect and may result from assuming that the tangent of the other acute angle is the negative of the reciprocal of the tangent of the angle described. Choice C is incorrect and may result from interpreting the tangent of the other acute angle as equal to the tangent of the angle described.