- #1
tonit
- 55
- 1
Homework Statement
Find 5th roots of unity solving algebraically x^5-1=0. Using the result, find sin18 and cos18
The Attempt at a Solution
[itex]
x^5 = 1\\
x = \sqrt[5]{1}[/itex]
since we have 5 roots:
[itex]
x_k, k = 0,1,2,3,4 \\ \\
x_k = e^{i\frac{2k\pi}{n}}, n=5 \\ x_0 = e^{i0} = 1\\ x_1 = e^{i\frac{2\pi}{5}} = cos\frac{2\pi}{5} + isin\frac{2\pi}{5} \\ x_2 = e^{i\frac{4\pi}{5}} = cos\frac{4\pi}{5} + isin\frac{4\pi}{5} \\ x_3 = e^{i\frac{6\pi}{5}} = cos\frac{6\pi}{5} + isin\frac{6\pi}{5} \\ x_5 = e^{i\frac{8\pi}{5}} = cos\frac{8\pi}{5} + isin\frac{8\pi}{5}[/itex]
now how do I find sin18 and cos18??