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Math Help - Solve the trigonometric equation

  1. #1
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    Solve the trigonometric equation

    Solve for x, y: 25^{\sin^2x+2\cos y +1}+25^{\cos^2y+2\sin x +1}=2.
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  2. #2
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    Quote Originally Posted by james_bond View Post
    Solve for x, y: 25^{\sin^2x+2\cos y +1}+25^{\cos^2y+2\sin x +1}=2.
    One possibility is that both those powers of 25 be equal to 1 so that 1+ 1= 2. In that case we must have sin^2x+ 2 cos y+ 1= 0 and cos^2y+ 2\sin x+ 1= 0. Adding those 2 equations, 1+ 2(cos x+ cos y)+ 2= 0 or cos x+ cos y= -3/2. From that cos y= -3/2- cos x. You can put that into cos^2 x+ cos^2 y= 1 to get a quadratic equation for cos x and then find x and y.
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  3. #3
    MHF Contributor red_dog's Avatar
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    \displaystyle 25^{\sin^2x+2\cos y+1}+25^{\cos^2y+2\sin x+1}\geq 2\sqrt{25^{(\sin x+1)^2+(\cos x+1)^2}}\geq 2

    The equality stands if 25^{\sin^2x+2\cos y+1}=25^{\cos^2y+2\sin x+1}=1\Rightarrow

    \left\{\begin{array}{ll}\sin^2x+2\cos y+1=0\\\cos^2y+2\sin x+1=0\end{array}\right.

    Adding the equations we have

    (\sin x+1)^2+(\cos y+1)^2=0\Rightarrow \sin x=\cos y=-1

    Now find x and y.
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