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A trig equation

  1. Mar 28, 2007 #1
    1. The problem statement, all variables and given/known data
    Find all solutions of the equation (tanx + sqrt(3))(cosx+2)=0


    2. Relevant equations
    Any of the identities


    3. The attempt at a solution

    (tanx + sqrt(3))(cosx + 2)=0
    tan(x)cos(x)+2tan(x) + sqrt(3)cos(x) + 2sqrt(3) = 0
    (sin(x)/cos(x))cos(x) + 2tan(x) + sqrt(3)cos(x) + 2sqrt(3) = 0
    sin(x) + 2tan(x) + sqrt(3)cos(x) + 2 sqrt(3) = 0
    sin^2(x) + 4tan^2(x) + 3cos^2(x) + 12 = 0 (square)
    sin^2(x) + cos^2(x) + 2cos^2(x) + 4tan^2(x) + 12 = 0
    2cos^2(x) + 4tan^2(x) + 13 = 0
    cos^2(x) + 2tan^2(x) + 6.5 = 0 (Divide by 2)
    1 - sin^2(x) + 2tan^2(x) + 6.5 = 0 (sin^2(x) + cos^2(x) = 1; cos^2(x) = 1 - sin^2(x))
    -sin^2(x) + 2tan^2(x) + 7.5 = 0
    -1 + 2sin^2(x) + 7.5/sin^2(x) (Multiply by sin^2(x))
    2sin^2(x) + (6.5 + sin^2(x) + cos^2(x))/sin^2(x) -1 = 0
    2sin^2(x) + 6.5 - 1 + cos^2(x) = 0
    sin^2(x) + sin^2 (x) + cos^2(x) + 5.5 = 0
    sin^2(x) + 6.5 = 0
    sin(x) + sqrt(6.5) = 0
    sin(x) = -sqrt(6.5)

    The book gives an answer of -(pi/3) + k(pi), which I'd think is correct over my equations - I honestly don't know how to get to an answer because the book is horrid at teaching and I'm out of school for a bit so I can't ask the teacher for help.
     
  2. jcsd
  3. Mar 28, 2007 #2
    I didn't really look at your work too closely, but this way seems more obvious to me.

    (tan(x) + sqrt(3))(cos(x) + 2) = 0

    This is equation is factored, i.e. in the form you need to solve the problem.

    Therefore, this equation equals 0 when:
    1) tan(x) + sqrt(3) = 0
    or
    2) cos(x) + 2 = 0

    1) check values of 'x' for which
    tan(x) = -sqrt(3)
    is true
    *can you solve for x?*

    2) check values of 'x' for which
    cos(x) = -2
    is true
    *can you sovle for x?*
     
  4. Mar 28, 2007 #3
    ey body, u can do this pretty simply,

    (tanx + sqrt(3))(cosx+2)=0

    when this is zero?

    it is zero when (tanx + sqrt(3))=0 or (cosx+2)=0, or both of them are zero, so what u have to do is just solve these two equations now, and find the overall answer.

    sorry, chaotic allready stated this. i did not see it
     
  5. Mar 28, 2007 #4

    HallsofIvy

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    I guarentee that if you put "Solve (x- 3)(x- 5)= 0" on a basic algebra test, 1/3 of the students will multiply it out and use the quadratic formula to solve x2- 8x+ 15= 0!
     
  6. Mar 28, 2007 #5

    symbolipoint

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    Homework Helper
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    HallsofIvy,

    Sometimes people forget the simple stuff and use the less simple stuff; they may forget to recognize what they have already studied. Usually, the students still UNDERSTAND the simple stuff, but they just forgot that this simple stuff is all that is needed.
     
  7. Mar 28, 2007 #6
    That's what happened with me. Too used to doing complex equations that I forgot that there were stupidly simple ways of doing it.

    Thanks for all the help!
     
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