When did Quad meet Trig?

Geometry Level 3

If y y is a real number, then which of the following quadratic equations has roots csc ⁡ 2 y \csc^2 y and sec ⁡ 2 y \sec^2 y ?

x 2 − 5 x + 5 = 0 x^2-5x+5=0 x 2 − 3 x + 3 = 0 x^2-3x+3=0 None x 2 − 2 x + 2 = 0 x^2-2x+2=0

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Aniruddha Bagchi by Aniruddha Bagchi , 20, India

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1 solution

Robert Szafarczyk
Apr 26, 2018

x 2 − 2 x + 2 = 0 x^{2}-2x+2=0 and x 2 − 3 x + 3 = 0 x^2-3x+3=0 have no real solutions.

We know that csc ⁡ 2 y \csc^{2}y and sec ⁡ 2 y \sec^{2}y can take any positive value. x 2 − 5 x + 5 x^{2}-5x+5 has two real solutions 5 + 5 2 \frac {5+\sqrt {5}}{2} and 5 − 5 2 \frac {5-\sqrt {5}}{2} . Substituting 1 cos ⁡ 2 x = 5 + 5 2 \frac {1}{\cos^{2} x} = \frac {5+\sqrt {5}}{2} and solving for sec ⁡ 2 x \sec^{2} x ( using cos ⁡ 2 x = 1 − sin ⁡ 2 x \cos^{2} x = 1 - \sin^{2} x ) gives that sec ⁡ 2 x = 5 − 5 2 \sec^{2} x = \frac {5-\sqrt {5}}{2} which proves the answer.

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