CR MATH 25 - Rotation of Axes (6 pages)

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Rotation of Axes



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Rotation of Axes

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6
School:
College of the Redwoods
Course:
Math 25 - College Trigonometry
College Trigonometry Documents

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Math 25 Fall 2004 Rotation of Axes In sections 10 2 4 we found that every equation of the form 1 Ax2 Cy 2 Dx Ey F 0 with A and C not both 0 can be transformed by completing the square into a standard equation of a translated conic section Thus the graph of this equation is either a parabola ellipse or hyperbola with axes parallel to the x and y axes there is also the possibility that there is no graph or the graph is a degenerate conic a point a line or a pair of lines In this section we will discuss the equation of a conic section which is rotated by an angle so the axes are no longer parallel to the x and y axes To study rotated conics we first look at the relationship between the axes of the conic and the x and y axes If we label the axes of the conic by u and v then we can describe the conic in the usual way in the uv coordinate system with an equation of the form 2 au2 cv 2 du ev f 0 Label the axis in the first quadrant by u so that the u axis is obtained by rotating the x axis through an angle with 0 2 see Figure 1 Likewise the v axis is obtained by rotating the y axis through the angle Now if P is any point with usual polar coordinates r then the polar coordinates of P in the uv system are r again see Figure 1 Thus u r cos r cos cos sin sin r cos cos r sin sin x cos y sin 3 and v r sin r sin cos cos sin r sin cos r cos sin y cos x sin These two equations in 3 describe how to find the uv coordinates of a point from its xy coordinates y v P r u x Figure 1 Using a similar derivation if we start with the uv coordinates then the xy coordinates are given by the equations 4 x u cos v sin and y v cos u sin Now if we substitute the expressions for u and v from 3 into equation 2 we will end up with an equation of the form Ax2 Bxy Cy 2 Dx Ey F 0 5 In other words a mixed xy term has been introduced Example 1 Starting with the ellipse 4u2 9v 2 36 in the uv system we obtain equation 5 in the xy system as follows 4 x cos y sin 2 9 y cos x sin 2 36 4 x2 cos2 2xy cos sin y 2



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