How to calculate the value of sin 50°????????????????????????????????????????????? ???????????????

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.