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TRIG rotated axis proof

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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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