Question

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

Answer 1

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`