1125 APPENDIX B Rotation of Axes Let OP = r, and let a represent the angle made by OP and the x′-axis. Figure 1 suggests that the following hold true. cos1u + a2 = OA r = x r , sin1u + a2 = AP r = y r , cos a = OB r = x′ r , sin a = BP r = y′ r These four statements can be written as follows. x = r cos1u + a2, y = r sin1u + a2, x′ = r cos a, y′ = r sin a The trigonometric identity for the cosine of the sum of two angles gives the following equation. x = r cos1u + a2 x = r 1cos u cos a - sin u sin a2 x = 1r cos a2cos u - 1r sin a2sin u Distributive property x = x′ cos u - y′ sin u Substitute. In the same way, the identity for the sine of the sum of two angles gives y = x′ sin u + y′ cos u. Derivation of Rotation Equations If we begin with an xy-coordinate system having origin O and rotate the axes about O through an angle u, the new coordinate system is a rotation of the xy-system. Trigonometric identities can be used to obtain equations for converting the coordinates of a point from the xy-system to the rotated x′y′@system. Let P be any point other than the origin, with coordinates 1x, y2 in the xysystem and 1x′, y′2 in the x′y′-system. See Figure 1. B Rotation of Axes ■ Derivation of Rotation Equations ■ Application of a Rotation Equation x9 y9 x y (x9, y9) P(x, y) O A B r u a Figure 1 Rotation Equations If the rectangular coordinate axes are rotated about the origin through an angle u, and if the coordinates of a point P are 1x, y2 and 1x′, y′2 with respect to the xy-system and the x′y′-system, respectively, then the rotation equations are as follows. x =x′ cos U −y′ sin U and y =x′ sin U +y′ cos U LOOKING AHEAD TO CALCULUS Rotation of axes is a topic traditionally covered in calculus texts, in conjunction with parametric equations and polar coordinates. The coverage in calculus is typically the same as that seen in this section.
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