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Math Help - Pre-Calculus: Trigonometric forumlas and more

  1. #1
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    Pre-Calculus: Trigonometric forumlas and more

    No need for a full out walktrhough - a brief description of what I need to do will suffice (AKA: enough to get me rolling on the right track, with enough advice to ensure I can do it all)


    1. Show that the coordinates of point (x,y) rotated counter-clockwise by an angle of θ around the origin have the new coordinates (xcosθ - ysinθ, xsinθ + ycosθ). Draw a picture that clearly illustrates your derivation, and provide plenty of explanation. Hint: consider the distance from the origin to (x,y) as "d", and use the angle sum formulas.

    2. Use this result to find the equation of the hyperbola xy = 2 rotated 45░ counterclockwise. Compute the vertices, foci, and asymptotes of each of these hyperbolas.

    3. Derive the equation of the general conic
    Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
    when rotated counterclockwise through an angle
    θ. Hint: the coefficient of x^2 becomes (A cos^2θ + Bsinθcosθ + Csin^2θ)

    4. Show that rotating the equation through an angle of θ where tan 2θ = B/(A-C) will eliminate that pesky xy term.

    5. For each of the following, find:
    -the angle of rotation that eliminates the xy term
    -the equation resulting from that rotation
    -the vertices, foci, and, if appropriate, asymptotes of each equation

    a. x^2 + 4xy + y^2 - 3 = 0
    b. 3x^2 - 10xy + 3y^2 - 32 = 0
    c. x^2 + 4xy + 4y^2 + 5(root5y) + 5 = 0

    I know it's lengthy and pretty annoying, but any help at all, no matter how little, is greatly appreciated. Thanks in advance!
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  2. #2
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    Quote Originally Posted by Sconts View Post
    1. Show that the coordinates of point (x,y) rotated counter-clockwise by an angle of θ around the origin have the new coordinates (xcosθ - ysinθ, xsinθ + ycosθ). Draw a picture that clearly illustrates your derivation, and provide plenty of explanation. Hint: consider the distance from the origin to (x,y) as "d", and use the angle sum formulas.
    ...
    Hello,

    I've added a screenshot of the complete work. Sorry!

    EB
    Attached Thumbnails Attached Thumbnails Pre-Calculus: Trigonometric forumlas and more-rotat_abbglg.gif  
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  3. #3
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    Hello, Sconts!

    I can help you with #1 . . .


    1. Show that the coordinates of point (x,y) rotated counter-clockwise by an angle of θ
    around the origin have the new coordinates (xĚcosθ - yĚsinθ, xĚsinθ + yĚcosθ).
    Draw a picture that clearly illustrates your derivation, and provide plenty of explanation.
    Hint: consider the distance from the origin to (x,y) as "d", and use the angle-sum formulas.
    Code:
            |                 P'
            |                 *(x',y')
            |               * :
            |             *   :
            |           *     :
            |      d  *       :     P
            |       *         :     *(x,y)
            |     *     d     *     :
            |   * θ     *     :     :
            | *   *  φ        :     :
          - * - - - - - - - - + - - + - X
            O

    Point P(x,y) make angle φ with the x-axis.
    Point P'(x',y') makes angle P'OP = θ.
    Let d = OP = OP'.

    The coordinates of P are: .x = dĚcosφ, .y = dĚsinφ .[1]

    The coordinates of P' are: .x' = dĚcos(θ + φ), .y' = dĚsin(θ + φ)


    We have: .x' .= .dĚcos(θ + φ) .= .dĚcosφĚcosθ - dĚsinφĚsinθ
    . . . . . . . . . . . . . . . . . . . . . . . . . . . . - . - . . .
    Substitute [1]: . . . . . . . . .x' .= . .x Ě cosθ . .- . .y Ě sinθ

    . . Therefore: .x' .= .xĚcosθ - yĚsinθ


    We have: .y' .= .dĚsin(θ + φ) .= .dĚcosφĚsinθ + dĚsinφĚcosθ
    . . . . . . . . . . . . . . . . . . . . . . . . . . - . - . - . - .
    Substitute [1]: . . . . . . . . .y' .= . .x Ě sinθ . .+ . .y Ě cosθ

    . . Therefore: .y' .= .xĚsinθ + yĚcosθ

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