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Math Help - Simultaneous equations

  1. #1
    pkr
    pkr is offline
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    Simultaneous equations

    cos(x)-sin(x-y) = 0
    -sin(y) + sin(x-y)=0

    however the solutions must be between 0 and pi/2, after a long while messing with it i got y=pi/6 and x=pi/3...can anyone show me how to actually calculate this thought, as obviously my way is pretty pointless.
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  2. #2
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    Hello, pkr!

    This is a tricky one, but I found a way . . .


    \begin{array}{cccc}{\color{blue}(1)} & \cos(x)-\sin(x-y) &=& 0 \\ {\color{blue}(2)} & \text{-}\sin(y) + \sin(x-y)&=&0 \end{array} \qquad 0 \,\leq \,x,y \,\leq \frac{\pi}{2}

    We have: . \begin{array}{ccccc} \cos x &=& \sin(x-y) \\ \sin y &=& \sin(x-y)\end{array} \quad\Rightarrow\quad \cos x \:=\:\sin y

    . . Hence, x\text{ and }y are complementary: . y \:=\:\tfrac{\pi}{2} - x .[3]


    Substitute into (1): . \cos x - \sin\left(x - [\tfrac{\pi}{2} - x]\right) \:=\:0 \quad\Rightarrow\quad\cos x - \sin(2x-\tfrac{\pi}{2}) \:=\:0

    . . \cos x - \bigg[\sin2x\cos\tfrac{\pi}{2} - \sin\tfrac{\pi}{2}\cos2x\bigg] \:=\:0 \quad\Rightarrow\quad\cos x  + \cos2x \:=\:0

    . . \cos x + 2\cos^2\!x - 1 \:=\:0\quad\Rightarrow\quad 2\cos^2\!x + \cos x - 1 \:=\:0


    This factors: . (\cos x + 1)(2\cos x - 1) \:=\:0


    And we have: . \cos x + 1 \;=\:0 \quad\Rightarrow\quad\cos x \:=\:-1 \quad\Rightarrow\quad{\color{red}\rlap{//////}} x \:=\:\pi

    . . and: . 2\cos x - 1\:=\:0 \quad\Rightarrow\quad \cos x \:=\:\tfrac{1}{2} \quad\Rightarrow\quad \boxed{x\:=\:\frac{\pi}{3}}

    Substitute into [3]: . y \:=\:\tfrac{\pi}{2} - \tfrac{\pi}{3} \quad\Rightarrow\quad \boxed{y\:=\:\frac{\pi}{6}}

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