Stitz-Zeager_College_Algebra_e-book

12 the goal of this exercise is to use vectors to

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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(...
View Full Document

Ask a homework question - tutors are online