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Trigonometric Equations Solution

  1. Sep 17, 2012 #1

    Peter G.

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    Hi,

    I had to solve without the aid of a calculator:

    2cos(x) = sin(2x)

    What I did was perform the following substitution:

    2cos(x) = 2sin(x)cos(x)

    I then cancelled and got:

    sin (x) = 1

    For 0 to 3pi,

    I got two answers, pi/2 and 5pi/2.

    I did not get one of the answers. Why?

    What did I do wrong?

    Thanks!
     
    Peter G., Sep 17, 2012
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  3. Sep 17, 2012 #2

    SammyS

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    2cos(x) = 2sin(x)cos(x) is also true if cos(x) = 0 , (in which case, you divided by zero when you cancelled.)

    A safer way:

    subtract 2cos(x) from both sides:

    0 = 2sin(x)cos(x) - cos(x) .

    Factor & use the zero product property of multiplication.
     
    SammyS, Sep 17, 2012
  4. Sep 17, 2012 #3

    Peter G.

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    Ok, thanks!
     
    Peter G., Sep 17, 2012
  5. Sep 17, 2012 #4

    Mark44

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    Staff: Mentor

    Just to emphasize what Sammy said, when you're solving equations, it's not a good idea to "cancel" since there is the chance that you will be losing a solution (just like you did here).

    Here's a simple example showing why cancelling is not a good idea:

    Solve for x in the equation x2 = 4x

    First attempt:
    Cancel x from each side to get x = 4.
    x = 4 is a solution, but the problem is, there is another that was lost by the cancel operation.

    Second attempt:
    Rewrite the equation as x2 - 4x = 0
    Factor to get x(x - 4) = 0
    Solution: x = 0 or x = 4
     
    Mark44, Sep 17, 2012
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