I am trying to simplify
cos(phi) + cos(phi - 2*pi/3)*e^(I*2*pi/3) + cos(phi - 4*pi/3)*e^(I*4*pi/3)
which I know reduces down to 1.5e^(I*phi)
I cannot get SymPy to recognize this. I have tried simplify, trigsimp, expand, etc. But nothing seems to work. Any suggestions?
Here is my code:
import numpy as np
%matplotlib inline
import matplotlib.pyplot as plt
import sympy as sp
from sympy import I
sp.init_printing()
phi = sp.symbols('phi', real = True)
vec = sp.cos(phi) + sp.cos(phi - 2*sp.pi/3)*sp.exp(I*2*sp.pi/3) + sp.cos(phi - 4*sp.pi/3)*sp.exp(I*4*sp.pi/3)
vec.simplify()
vec.rewrite(sp.exp).simplify()
vec.rewrite(sp.exp).expand().simplify()
None of these produce the expected result.
I can confirm my result manually, by substituting values in for phi like this:
sp.simplify(vec.rewrite(sp.exp).simplify() - 3/2*sp.exp(I*phi)).evalf(subs={phi:3})
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Answer
It’s not obvious but you can get there like this:
In [40]: phi = symbols('phi', real=True)
In [41]: e = cos(phi) + cos(phi - 2*pi/3)*E**(I*2*pi/3) + cos(phi - 4*pi/3)*E**(I*4*pi/3)
In [42]: e
Out[42]:
-2⋅ⅈ⋅π 2⋅ⅈ⋅π
─────── ─────
3 ⎛ π⎞ 3 ⎛ π⎞
- ℯ ⋅sin⎜φ + ─⎟ + cos(φ) - ℯ ⋅cos⎜φ + ─⎟
⎝ 6⎠ ⎝ 3⎠
In [43]: e.rewrite(exp).expand().rewrite(sin).expand().rewrite(exp)
Out[43]:
ⅈ⋅φ
3⋅ℯ
──────
2