12 the goal of this exercise is to use vectors to

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Unformatted text preview: apter 7. It is natural to wonder if we can always do this. That is, given an equation of the form Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0, with B = 0, is there an angle θ so that if we rotate the x and y axes counter-clockwise through that angle θ, the equation in the rotated variables x and y contains no x y term? To explore this conjecture, we make the usual substitutions x = x cos(θ) − y sin(θ) and y = x sin(θ) + y cos(θ) into the equation Ax2 + Bxy + Cy 2 + Dx + Ey + F = 0 and set the coefficient of the x y term equal to 0. Terms containing x y in this expression will come from the first three terms of the equation: Ax2 , Bxy and Cy 2 . We leave it to the reader to verify that x2 = (x )2 cos2 (θ) − 2x y cos(θ) sin(θ) + (y )2 sin(θ) xy = (x )2 cos(θ) sin(θ) + x y cos2 (θ) − sin2 (θ) − (y )2 cos(θ) sin(θ) y 2 = (x )2 sin2 (θ) + 2x y cos(θ) sin(θ) + (y )2 cos2 (θ) 830 Applications of Trigonometry The contribution to the x y -term from Ax2 is −2A cos(θ) sin(θ), from Bxy it is B cos2 (θ) − sin2 (θ) , and from Cy 2 it is 2C cos(θ) sin(θ). Equating the x y -term to 0, we get −2A cos(θ) sin(θ) + B cos2 (θ) − sin2 (θ) + 2C cos(θ) sin(...
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