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

1 Answer
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#