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Thread: Two geometry equations

  1. #1
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    Two geometry equations

    How ?
    Attached Thumbnails Attached Thumbnails Two geometry equations-trigo.jpg  
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  2. #2
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    For Problem 1:
    Factor to get, call $\displaystyle y=x-\frac{\pi}{5}$
    $\displaystyle \cos y (2\cos^2y-3\cos y-1)=0$
    Thus,
    $\displaystyle \cos y=0$ or $\displaystyle 2\cos^2y-3\cos y-1=0$
    Solving the second equation with quadradic formula:
    $\displaystyle \cos y=\frac{3\pm\sqrt{17}}{4}$
    Thus,
    $\displaystyle \cos y=0$
    $\displaystyle \cos y\approx 1.78$--Impossible.
    $\displaystyle \cos y\approx -.28$
    Solve for $\displaystyle y$
    after that you can solve for $\displaystyle x$
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  3. #3
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    The product formula,
    $\displaystyle \sin x\sin y=\frac{1}{2}[\cos(x-y)-\cos(x+y)]$
    Thus,
    $\displaystyle \sin (\pi/4+x)\sin (\pi/4-x)$
    Can be expressed as,
    $\displaystyle \frac{1}{2}[\cos(2x)-\cos(\pi/2)]$
    But $\displaystyle \cos(\pi/2)=0$ thus,
    $\displaystyle \frac{1}{2}\cos 2x=.33$
    Thus,
    $\displaystyle \cos 2x=.66$
    Now you can solve for $\displaystyle 2x$ for which you can solve for $\displaystyle x$
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