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rotated axis

Trig

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Trigonometry
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conics
rotated axis

10.5 Notes - Rotation of Axes At times we may be faced with an equation of a conic section whose axes would not be parallel to the x or y-axes. (These conics have an equation of the form where .) As a result it may be difficult to ascertain the nature of the graph or its position on the coordinate system. In this case we may simplify the equation by a process of rotating the axes. ? In the figure the axes have been rotated through an acute angle about the origin to produce a new pair of axes, which we will call the X and Y axes. A point P that has coordinates (x, y) in the old system has coordinates (X,Y) in the new system. Let r equal the distance of P from the origin. Let be the angle that segment OP makes with the X axis. ? ?

TRIG rotated axis proof

Subject: 
Trigonometry
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conics
rotated axis

10.5B Notes ? Rotation of Axes The general equation of the second degree in two variables may be written where A, B, C, D, E and F are real coefficients and not all of A, B or C are zero. We can transform this equation into an equation in terms of X and Y by rotating the axes through an appropriate angle ?. To find the angle that works substitute for x and y using the rotation formulas. x = X cos ? ? Y sin ?, y = X sin ? + Y cos ? in terms of X and Y is Expanding this and collecting like terms, (which is quite a job), we obtain an equation of the form ?.. Where In order to eliminate the XY-term, we need to choose so that . That is or equivalently . Note: Don?t forget your half angle formulas for sine and cosine. ?
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