csc ((2pi)/9) = 1/sin ((2pi)/9)csc(2π9)=1sin(2π9). Call sin ((2pi)/9) = xsin(2π9)=x
Use the trig identity:
sin 3x = 3sin x - 4sin^3 xsin3x=3sinx−4sin3x
sin ((6pi)/9) = sin ((2pi)/3) = sqrt3/2sin(6π9)=sin(2π3)=√32
sqrt3/2 = 3x - 4x^3√32=3x−4x3
4x^3 - 3x + sqrt3/2 = 04x3−3x+√32=0
Solve this cubic equation by graphing calculator to get x.
x = sin ((2pi)/9) = sin 40^@x=sin(2π9)=sin40∘
graph{4x^3 - 3x + sqrt3/2 [-1.25, 1.25, -0.625, 0.625]}
By estimation, we get:
sin x1 = 0.33 --> x1 = 19^@27x1=19∘27 --> (Rejected)
sin x2 = 0.64 --> x2 = 39.79 = 40^@ x2=39.79=40∘OK
sin x3 = - 0.98 --> x3 = -78^@52x3=−78∘52 --> (Rejected)
Finally
sin x = sin ((2pi)/9) = sin 40^@ = 0.64sinx=sin(2π9)=sin40∘=0.64->
csc ((2pi)/9) = 1/(0.64) = 1.56csc(2π9)=10.64=1.56
