Recently, I learned that cos(π/7) and sin(π/7) cannot be written in terms of algebraic numbers. This is pretty shocking.
The half-angle formulas
cos2(x2)=1+cos(x)2,sin2(x2)=1−cos(x)2
can be used to find expressions for fractions of π with powers of two as denominators.
You begin with
cos(π)=−1,sin(π)=0
Using the half-angle formulas gives
cos(π2)=0,sin(π2)=1
The idea is to repeat this procedure to get other fractions.
Using the half-angle formulas gives
cos(π4)=√22,sin(π4)=√22
Using the half-angle formulas gives
cos(π8)=√2+√22,sin(π8)=√2−√22
Using the half-angle formulas gives
cos(π16)=√2+√2+√22,sin(π16)=√2−√2+√22
Using the half-angle formulas gives
cos(π32)=√2+√2+√2+√22,sin(π32)=√2−√2+√2+√22
You can already see the pattern. As the denominator grows, the cosine tends to 1, and the sine tends to 0.
Let Rk be the following unitary matrix
Rk=[100exp(−2πi2k)]
Note that
R19=[1000.9999999999−0.00001198422491i]≈[1001]
So after k=19, the matrix Rk can be approximated by the identity.