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Math Help - Solving some equations

  1. #1
    Senior Member
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    Solving some equations

    Hello.

    Does anyone know how to solve these equations:

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

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

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

    Help would be much appreciated.

    Rapha
    Last edited by Rapha; February 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?

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

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

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


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

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

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


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

    . . And: . \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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