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Dear student
sin50=sin(90–40)
sin(90–40)=cos(40)….(we take the complementary trigonometric function if angle is in the form of n{π/2}±phi)
cos40=cosx
cos120=cos3x
cos3x=cos2xcosx-sin2xsinx
cos3x={cos²x-sin²x}(cosx)-2sinxcosx(sinx)
cos3x=cos³x-sin²xcosx-2sin²xcosx
cos3x=cos³x-3sin²xcosx
cos3x=cos³x-3cosx+3cos³x
cos3x=4cos³x-3cosx
cos 3x=cos 120=-cos 60=-1\2
8cos³x-6cosx+1=0
And by the courtesy of desmos graphing calculator, we have been able to obtain one negative and two positive solutions for cosx
There can't be a negative solution since all cosines are positive in quadrant 1,
sine function increases from 0 to 90, therefore sin50>sin45, and sin50>1/√2
So according to the next two conditions, 0.174 can't be a solution, and hence the only remaining solution is 0.766, which makes it our answer.
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