Find real part of \(\displaystyle{\frac{{{1}}}{{{1}-{e}^{{{i}\frac{\pi}{{7}}}}}}}\) How can you

Annie Rice

Annie Rice

Answered question

2022-04-12

Find real part of 11eiπ7
How can you find
Re(11eiπ7)
I put it into wolframalpha and got 12, but I have no idea where to begin. I though maybe we could use the fact that
1z=z|z|2,
where z is the conjugate of z. Unfortunately, the magnitude doesn't seem to be a nice number. I feel like this might be a trigonometry question in disguise, but converting eiπ7=cos(π7)+isin(π7) hasn't been very fruitful.

Answer & Explanation

strasnihwhge

strasnihwhge

Beginner2022-04-13Added 14 answers

Let
z=11eit
where t is real and eit1. Then
z+z¯=11-eit+11-e-it=11-eit+eiteit-1=1-eit1-eit=1
Therefore the real part of z equals 12

Drantumcem0

Drantumcem0

Beginner2022-04-14Added 10 answers

11eiπ7=eiπ14eiπ14eiπ14=
=12ieiπ14sinπ14=
=12+i2cosπ14sinπ14=12+i2cotπ14

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