$\"\"$

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Rotation formula :

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If the $\"\"$ and $\"\"$ - axes are rotated through an angle $\"\"$ , the coordinates $\"\"$ of

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a point $\"\"$ relative to the $\"\"$ - plane and the coordinates $\"\"$ of the same point

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relative to the new $\"\"$ and $\"\"$ - axis and are related by the formulas

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$\"\"$ and $\"\"$ .

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The general form is $\"\"$

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The angle is $\"\"$ .

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$\"\"$

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The equation is $\"\"$ .

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Compare $\"\"$ with $\"\"$

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$\"\"$ and $\"\"$ .

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The angle is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$

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$\"\"$

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$\"\"$

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Rotation of $\"\"$ -axis $\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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Rotation of $\"\"$ -axis $\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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The rotation formulas are $\"\"$ and $\"\"$ .

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$\"\"$

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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Divide each side 12.

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$\"\"$

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The above equation represents an ellipse.

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$\"\"$

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Graph :

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(1) Draw the coordinate plane.

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(2) Draw the rotated coordinate plane

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(3)Graph the function $\"\"$ .

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$\"\"$

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$\"\"$

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The equation of an ellipse is $\"\"$ .

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The graph the function $\"\"$ .

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$\"\"$