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Thread: Simultaneous equations involving cos and sin

  1. #1
    Junior Member CuriosityCabinet's Avatar
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    Simultaneous equations involving cos and sin

    $\displaystyle (cos \theta)x - (sin \theta)y=2$
    $\displaystyle (sin\theta)x + (cos \theta)y=1$

    In the range $\displaystyle 0 \leq \theta < 2 \pi$ is it solvable? Or are there values of theta when the equations are not solvable?
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  2. #2
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    Quote Originally Posted by CuriosityCabinet View Post
    $\displaystyle (cos \theta)x - (sin \theta)y=2$
    $\displaystyle (sin\theta)x + (cos \theta)y=1$

    In the range $\displaystyle 0 \leq \theta < 2 \pi$ is it solvable? Or are there values of theta when the equations are not solvable?
    Assuming you mean to solve for x and y, just treat $\displaystyle sin(\theta)$ and $\displaystyle cos(\theta)$ as constants. In particular, you can multiply the first equation by $\displaystyle cos(\theta)$, the second equation by $\displaystyle sin(\theta)$, and add the equations to eliminate y. A system of equations would be "non-solvable" if and only if the resulting coefficient of x were 0. But a rather nice thing happens here!
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