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The equation sin x = 1 + cos^2x has to be solved.
sin x = 1 + cos^2x
Use the relation sin^2x + cos^2 x = 1 or cos^2x = 1 - sin^2x
The given equation can be written as:
sin x = 1 + 1 - sin^2x
sin^2x + sin x - 2 = 0
sin^2x + 2sinx - sin x - 2 = 0
sin x(sin x + 2) - 1(sin x + 2) = 0
(sin x - 1)(sin x + 2) = 0
sin x = 1 and sin x = 2
But the value of sinusoidal functions sin x and cosine x lies in the set [-1, 1]
Therefore the root sin x = 2 can be ignored.
sin x = 1 gives x = 90 degrees.
To solve the equation sinx=1+cos^2x, we use the trigonometrical identiy sin^2+cos^2=1
From the above identity, cos^2x =1-sin^2x.Replacing this fact in the given equation we get:
sinx=1+(1-sin^2x). Rearrange this as a quadratic in sinx we get,
sinx+1/2 =sqrt(2.25) =+1.5 or -1.5
sinx=1.5-0.5 =1 or sinx =-1.5-0.5=-2.0 is not feasible, as sinx is always obeying the inequality, -1<=sinx<=1.
Thus sinx =1 gives x=Pi/2 is the only practical solution or x=Pi/2+2nPi is the general solution.
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