How do you evaluate csc(2pi/9)?

1 Answer
Nghi N.
Oct 2, 2015

Find csc((2pi)/9)

Ans: 1.56

Explanation:

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

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