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Math Help - Trigo-system

  1. #1
    Super Member dhiab's Avatar
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    Trigo-system

    Solve : sin(x)sin(y)=\frac{\sqrt{3}}{4}
    cos(x)cos(y)=\frac{\sqrt{3}}{4}
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    Quote Originally Posted by dhiab View Post
    Solve : sin(x)sin(y)=\frac{\sqrt{3}}{4}
    cos(x)cos(y)=\frac{\sqrt{3}}{4}
    Dear dhiab,

    Use the trignometric identities given below. Then you would be able to solve your problem.

    cos(A+B)=cosAcosB-sinAsinB

    cos(A-B)=cosAcosB+sinAsinB

    Hope this will help you.
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  3. #3
    Super Member Failure's Avatar
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    Quote Originally Posted by dhiab View Post
    Solve : sin(x)sin(y)=\frac{\sqrt{3}}{4}
    cos(x)cos(y)=\frac{\sqrt{3}}{4}
    It follows that:
    \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)=\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{4}=0

    Hence,

    x+y=\frac{\pi}{2}+n\pi, \qquad n\in\mathbb{Z}

    Now plug y=\frac{\pi}{2}+n\pi into one of the above equations to learn what additional conditions x and y have to satisfy...
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  4. #4
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    Quote Originally Posted by dhiab View Post
    Solve : sin(x)sin(y)=\frac{\sqrt{3}}{4}
    cos(x)cos(y)=\frac{\sqrt{3}}{4}
    cos(x)cos(y) + sin(x)sin(y) = \frac{\sqrt{3}}{2}]

    cos(x-y) = \frac{\sqrt{3}}{2}

    x - y = π/6 .......(1)

    If you subtract the above two equation you will get

    cos(x+y) = 0 of x+y = π/2.....(2)

    From eq.1 and 2, find x and y.
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  5. #5
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    Hello dhiab
    Quote Originally Posted by dhiab View Post
    Solve : sin(x)sin(y)=\frac{\sqrt{3}}{4}
    cos(x)cos(y)=\frac{\sqrt{3}}{4}
    Using \sin x \sin y = \tfrac12\big(\cos(x-y) -\cos(x+y)\big):
    \sin x\sin y=\frac{\sqrt{3}}{4}

    \Rightarrow \cos(x-y)-\cos(x+y) = \frac{\sqrt3}{2} ... (1)
    Similarly:

    \cos x \cos y =\frac{\sqrt{3}}{4}


    \Rightarrow \cos(x-y)+\cos(x+y) = \frac{\sqrt3}{2} ... (2)

    Add (1) and (2):
    \cos(x-y) =\frac{\sqrt3}{2}

    \Rightarrow x-y = 2n\pi\pm\frac{\pi}{6} ... (3)

    Subtract (1) and (2):
    \cos(x+y)=0

    \Rightarrow x+y = 2n\pi\pm\frac{\pi}{2} ...(4)

    Add (3) and (4):
    x=2n\pi \pm \frac{\pi}{3}

    \Rightarrow y = 2n\pi\pm\frac{\pi}{6}
    Grandad
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