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Thread: Solving some equations

  1. #1
    Senior Member
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    Solving some equations

    Hello.

    Does anyone know how to solve these equations:

    $\displaystyle X*cos(t*a) + Y*sin(t*a) = 0 $

    $\displaystyle X*cos(t*b) + Y*sin(t*b) = 0 $

    I want to find nontrivial solutions for X,Y and t. I only know that $\displaystyle a \not= 0 \not= b$. Furthermore $\displaystyle a,b \in \mathbb{R} $

    Help would be much appreciated.

    Rapha
    Last edited by Rapha; Feb 24th 2010 at 06:21 AM.
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  2. #2
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    Hello, Rapha!

    Does anyone know how to solve these equations?

    . . $\displaystyle \begin{array}{ccccc}x\cos(at) + y\sin(at) &=& 0 & [1] \\
    x\cos(bt) + y\sin(bt) &=& 0 & [2] \end{array} $

    I want to find nontrivial solutions for $\displaystyle x,y$ and $\displaystyle t.$
    I only know that $\displaystyle a,b \neq 0$. .Furthermore: $\displaystyle a,b \in \mathbb{R} $

    $\displaystyle \begin{array}{ccccccc}
    \text{Multiply [1] by }\sin(bt)\!: & x\cos(at)\sin(bt) + y\sin(at)\sin(bt) &=& 0 & [3] \\
    \text{Multiply [2] by }\sin(at)\!: & x\sin(at)\cos(bt) + y\sin(at)\sin(bt) &=& 0 & [4] \end{array}$


    Subtract [4] - [3]: .$\displaystyle x\sin(at)\cos(bt) - x\cos(at)\sin(bt) \;=\;0 $

    . . . . . . . . . . . . . $\displaystyle x\bigg[\sin(at)\cos(bt) - \cos(at)\sin(bt)\bigg] \;=\;0 $

    . . . . . . . . . . . . . . . . . . . . . . . . . .$\displaystyle x\sin(at - bt) \;=\;0$


    We have: .$\displaystyle x \:=\:0 \quad \hdots \text{ trivial solution}$

    . . And: .$\displaystyle \sin(a-b)t \;=\;0 \quad\Rightarrow\quad (a-b)t \;=\;\pi n \quad\Rightarrow\quad t \;=\;\frac{\pi n}{a-b}$

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  3. #3
    Senior Member
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    Hello Soroban.

    That was really helpful, thank you very much!

    Kind regards
    Rapha
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