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Math Help - solving complex equation using euler's formula

  1. #1
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    solving complex equation using euler's formula

    Not sure where this belonged in the forum, but hopefully I got close.

    I need to find values for c and x, that are both real and positive that satisfy this equation. Im pretty sure that Euler's formula will be of use, I cant figure out what the first step would be. Thanks.

    cos(4t-1) - 2sin(4t+2) = c cos(4t+x)
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  2. #2
    Grand Panjandrum
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    Re: solving complex equation using euler's formula

    Quote Originally Posted by snaes View Post
    Not sure where this belonged in the forum, but hopefully I got close.

    I need to find values for c and x, that are both real and positive that satisfy this equation. Im pretty sure that Euler's formula will be of use, I cant figure out what the first step would be. Thanks.

    cos(4t-1) - 2sin(4t+2) = c cos(4t+x)
    This requires the application of trig identities to the left hand side to reduce it to the form of the left hand side.

    What have you tried?

    CB
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  3. #3
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    Re: solving complex equation using euler's formula

    I tried splitting these up with trig identities before and got:

    cos(1)cos(4t) + sin(1)sin(4t) - 2cos(4t)sin(2) + 2cos(2)sin(4t)

    Next, I factored out the constants. getting:

    [cos(1)-2sin(2)]cos(4t) + [sin(1)+2cos(2)]sin(4t).

    Now I just gotta find a way to combine these and just get a cosine term on the left side so that it will match the right.
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  4. #4
    MHF Contributor chisigma's Avatar
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    Re: solving complex equation using euler's formula

    Quote Originally Posted by snaes View Post
    I tried splitting these up with trig identities before and got:

    cos(1)cos(4t) + sin(1)sin(4t) - 2cos(4t)sin(2) + 2cos(2)sin(4t)

    Next, I factored out the constants. getting:

    [cos(1)-2sin(2)]cos(4t) + [sin(1)+2cos(2)]sin(4t).

    Now I just gotta find a way to combine these and just get a cosine term on the left side so that it will match the right.
    Your original idea to use Euler's formula seems to me pretty good!... remember that from the Euler's formula...

    e^{i x} = \cos x + i\ \sin x (1)

    ... You can derive the identities...

    \cos x = \frac{e^{i x}+e^{-i x}}{2} (2)

    \sin x = \frac{e^{i x}-e^{-i x}}{2i} (3)

    What I suggest to You is to write all sin and cos in Your equation in 'exponential form' and then to set, for example, e^{i t}=z...

    Kind regards
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  5. #5
    Grand Panjandrum
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    Re: solving complex equation using euler's formula

    Quote Originally Posted by snaes View Post
    I tried splitting these up with trig identities before and got:

    cos(1)cos(4t) + sin(1)sin(4t) - 2cos(4t)sin(2) + 2cos(2)sin(4t)

    Next, I factored out the constants. getting:

    [cos(1)-2sin(2)]cos(4t) + [sin(1)+2cos(2)]sin(4t).

    Now I just gotta find a way to combine these and just get a cosine term on the left side so that it will match the right.
    Try putting

    \tan(\phi)=\frac{\sin(1)+2\cos(2)}{\cos(1)-2\sin(2)}

    and:

    c=\sqrt{[\sin(1)+2\cos(2)]^2+[\cos(1)-2\sin(2)]^2}

    CB
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  6. #6
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    Re: solving complex equation using euler's formula

    This worked great thanks! It just took me a while, because I messed up a negative sign for a while...

    Quote Originally Posted by CaptainBlack View Post
    Try putting

    \tan(\phi)=\frac{\sin(1)+2\cos(2)}{\cos(1)-2\sin(2)}

    and:

    c=\sqrt{[\sin(1)+2\cos(2)]^2+[\cos(1)-2\sin(2)]^2}

    CB
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  7. #7
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    Re: solving complex equation using euler's formula

    Just to follow up - I got close with this but couldnt finish it this way. I did get the answer I was looking for with the other route. Thanks!

    Quote Originally Posted by chisigma View Post
    Your original idea to use Euler's formula seems to me pretty good!... remember that from the Euler's formula...

    e^{i x} = \cos x + i\ \sin x (1)

    ... You can derive the identities...

    \cos x = \frac{e^{i x}+e^{-i x}}{2} (2)

    \sin x = \frac{e^{i x}-e^{-i x}}{2i} (3)

    What I suggest to You is to write all sin and cos in Your equation in 'exponential form' and then to set, for example, e^{i t}=z...

    Kind regards
    Follow Math Help Forum on Facebook and Google+

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