Find the number of solutions x in [0,2pi) to the equation 7sin x +2(cos x)^2 =5.
The problem seems to be hard as the equation involved both cosx and sinx But we can relate the two as (cosx)^2 + (sinx)^2 = 1, so we can rearrange this as 0 = 2(sinx)^2 - 7sinx + 3. This quadratic in sinx has solutions sinx = (7+5)/4 = 3 or sinx = (7-5)/4 = 0.5. As |sinx| is at most 1, we have sinx = 0.5, so that x = pi/6 or x = 5pi/6. (We should draw a graph of sinx to make sure we have all the solutions in the stated range.) So the answer is 2.
JR
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