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Math Help - Trig general solution

  1. #1
    Member srirahulan's Avatar
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    Trig general solution

    I have a risk at these two Trigonometric equation to find out the general solution.

    1.\ sin2x-cos2x-sinx+cosx=0\\ \  2.\ 4cos3x-sin2xcosx=0
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  2. #2
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    Re: Trig general solution

    Quote Originally Posted by srirahulan View Post
    I have a risk at these two Trigonometric equation to find out the general solution.
    1.\ sin2x-cos2x-sinx+cosx=0\\ \  2.\ 4cos3x-sin2xcosx=0
    Have a look at this.
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  3. #3
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Trig general solution

    Principal general solutions in the complex plane found as follows:

    Substitute sin() & cos() functions with their equivalents in complex plane:

    z=x+\text{i y}

     2 {Cos}({n$\theta $})=\left(z^n+\frac{1}{z^n}\right)

     2 i {Sin}({n$\theta $})=\left(z^n-\frac{1}{z^n}\right)

    to obtain:

    \frac{1}{2 i}\left(z^2-\frac{1}{z^2}\right)-\frac{1}{2}\left(z^2+\frac{1}{z^2}\right)-\frac{1}{2 i}\left(z-\frac{1}{z}\right)+\frac{1}{2}\left(z+\frac{1}{z} \right) =0

    Solve to obtain the 4 principal roots:

    \{z\to i\},\{z\to 1\},\left\{z\to \frac{1}{2} \left(-i-\sqrt{3}\right)\right\},\left\{z\to \frac{1}{2} \left(-i+\sqrt{3}\right)\right\}

    Similarly, the 2nd equation will become:

    2\left(z^3+\frac{1}{z^3}\right)-\frac{1}{2i}\left(z^2-\frac{1}{z^2}\right)*\frac{1}{2}\left(z+\frac{1}{z  }\right)=0

    with 6 principal roots:

    z=-i, z=i, z=-\frac{2+i}{\sqrt{5}}, z=\frac{2+i}{\sqrt{5}}, z=-\frac{3-2 i}{\sqrt{13}}, z=\frac{3-2 i}{\sqrt{13}}
    Last edited by MaxJasper; August 26th 2012 at 06:18 PM.
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  4. #4
    Member srirahulan's Avatar
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    Re: Trig general solution

    I cant understand your solution please give me the way to solve the equation and find the general solution.
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  5. #5
    Senior Member MaxJasper's Avatar
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    Re: Trig general solution

    Attached Thumbnails Attached Thumbnails Trig general solution-sin-2x-cos-2x-sin-x-cos-x-.png  
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  6. #6
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    Re: Trig general solution

    I'll try to explain somewhat what is going on. On the complex plane we have that

    \cos(nx) = \frac{e^{inx}+e^{-inx}}{2},\quad \sin(nx) = \frac{e^{inx}-e^{-inx}}{2i}

    If you let z = e^{ix} we get this representation

    \cos(nx) = \frac{z^n+z^{-n}}{2},\quad \sin(nx) = \frac{z^n-z^{-n}}{2i}

    Next, we substitute these in your equations. The first one is

    \frac{z^2-z^{-2}}{2i}-\frac{z^2+z^{-2}}{2}-\frac{z-z^{-1}}{2i}+\frac{z^1+z^{-1}}{2} = 0

    After we multiply everything by 2iz^2 and expand, the equation becomes

    -(1+i) + (1+i)z -(1-i) z^3 + (1-i)z^4 = 0

    The second equation becomes

    4\frac{z^3+z^{-3}}{2}-  \frac{z^2-z^{-2}}{2i}\frac{z+z^{-1}}{2}= 0

    After we multiply everything by 4iz^3 and expand, the equation becomes

    (1 + 8 i) + z^2 - z^4 - (1-8i) z^6 = 0

    Solving each of these equations gives you the roots that MaxJasper wrote (I used a math program to find them for me). These equations need to hold simultaneously, so the only roots you accept are ones that solve both equations. If you inspect them, you will see that only z = i is common to both. Now, we have that e^{ix} = i, so you are looking for solutions of this new equation

    e^{ix} = i

    Knowing a bit about the complex exponential, you will immediately see that the solutions to this new equation are

    x = \frac{\pi}{2} + 2\pi n, \quad n = 0, \pm 1, \pm 2, \dots

    As a final step you need check which of these solves the original equations, if any. This is indeed the case.
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