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Math Help - Trig equation problem using sum & difference identities

  1. #1
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    Trig equation problem using sum & difference identities

    Hi there,

    I'm looking for some help on two questions, I think I have the first one but would like confirmation that I'm correct. The second one I'm getting slightly confused with.

    thanks.

    Solve the following for 0 <= x <= pi

    (i) Sin 5x - sin 2x = 0

    (ii) Cos 5x + Sin 5x + Cos 3x + Sin 3x = 0

    Okay,

    For (i):

    I use the sum and difference identities and get:

    2cos (7x/2) * sin (3x/2) = 0

    Therefore when either of these equal zero the statement is true.

    Therefore,
    2cos (7x/2) = 0
    x = pi/2 as a general solution and calculate the next six.
    the divide by 7/2 to give x and select the ones in the correct range which are:

    pi/7, 3pi/7, 5pi/7 & pi

    Do the same for Sine (3x/2) = 0 and I get 0 & 2pi/3.

    These are all the solutions.

    Is this correct??

    (ii) It looks like this is a case of two sum and difference identities combined but I just get into a mess.

    Can anyone help???

    many thanks,

    D
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  2. #2
    MHF Contributor
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    Hi

    (i) correct

    (ii) Cos 5x + Sin 5x + Cos 3x + Sin 3x = 0
    Cos 5x + Cos 3x + Sin 5x + Sin 3x = 0
    2 Cos 4x * Cos x + 2 Sin 4x * Cos x = 0
    2 Cos x (Cos 4x + Sin 4x) = 0
    i let you finish the job
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  3. #3
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    Thanks for the post. How do i make (Cos 4x + sin 4x) = 0 ? i can't see how this would be done...
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  4. #4
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    Quote Originally Posted by dojo View Post
    Thanks for the post. How do i make (Cos 4x + sin 4x) = 0 ? i can't see how this would be done...
    Express it as r\sin(4x+\alpha).
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  5. #5
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    OK, so I get:

    ... 2 Cos x (Cos 4x + Sin 4x) = 0
    2 cos x = 0 therefore x = pi/2

    cos 4x + sin 4x = 0
    Therefore, r sin( 4x - alpha) = 0
    sin (4x - pi/4) = 0
    (4x - pi/4) = 0, pi, 2*pi, 3*pi ...
    4x = pi/4 , 5pi/4 , 9pi/4 , 13pi/4
    x = pi/16 , 5pi/16 , 9pi/16 , 13pi/16

    So as the range is 0 <= x <= pi
    the full answers are:
    pi/16 , 5pi/16 , 9pi/16 , 13pi/16 , pi/2

    Is this correct??

    thanks for all your help

    D
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