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

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

    $\displaystyle cos(A+B)=cosAcosB-sinAsinB$

    $\displaystyle cos(A-B)=cosAcosB+sinAsinB$

    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 : $\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$
    $\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} $
    It follows that:
    $\displaystyle \cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)=\frac{\sqrt{3}}{4}-\frac{\sqrt{3}}{4}=0$

    Hence,

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

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

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

    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 : $\displaystyle sin(x)sin(y)=\frac{\sqrt{3}}{4}$
    $\displaystyle cos(x)cos(y)=\frac{\sqrt{3}}{4} $
    Using $\displaystyle \sin x \sin y = \tfrac12\big(\cos(x-y) -\cos(x+y)\big)$:
    $\displaystyle \sin x\sin y=\frac{\sqrt{3}}{4}$

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

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


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

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

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

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

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

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

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