$\"\"$

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Rotation formula :

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If the $\"\"$ and $\"\"$ -axes are rotated through an angle $\"\"$ , the coordinates $\"\"$ of a point P relative to the $\"\"$ -plane and the coordinates $\"\"$ of the same point relative to the new $\"\"$ and $\"\"$ -axis and are related by the formulas $\"\"$ and $\"\"$ .

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The general form is $\"\"$

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The angle is $\"\"$ .

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If $\"\"$ , then $\"\"$ , so $\"\"$ .

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If $\"\"$ , then $\"\"$ , so $\"\"$ .

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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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Since $\"\"$ , the angle lies in second quadrant.

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$\"\"$

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$\"\"$ .

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Use Pythagorean theorem :

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$\"\"$ .

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$\"\"$

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$\"\"$ .

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In second quadrant cosine function is negative.

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$\"\"$ .

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$\"\"$

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Half angle formula of sine function is $\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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$\"\"$

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Half angle formula of cosine function is $\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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$\"\"$

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$\"\"$

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Rotation of $\"\"$ -axis :

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ in above equation.

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$\"\"$

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Rotation of $\"\"$ -axis :

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$\"\"$ .

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Substitute $\"\"$ and $\"\"$ 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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The above equation is a parabola.

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The general form of parabolic equation is $\"\"$ .

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Where $\"\"$ is the vertex and $\"\"$ is focus.

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

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$\"\"$ .

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$\"\"$

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The vertex is $\"\"$ .

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The focus is $\"\"$ .

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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 of the function $\"\"$ .

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$\"\"$

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$\"\"$

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The angle is $\"\"$

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The function $\"\"$ .

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The vertex is $\"\"$ .

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The focus is $\"\"$ .

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The graph of the function $\"\"$

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$\"\"$ .