Sum of Cosines of 2 k pi over 5

Theorem

\(\ds \)

\(\)

\(\ds 1 + \cos \dfrac {2 \pi} 5 + \cos \dfrac {4 \pi} 5 + \cos \dfrac {6 \pi} 5 + \cos \dfrac {8 \pi} 5\)

\(\ds \)

\(=\)

\(\ds 1 + \cos 72 \degrees + \cos 144 \degrees + \cos 216 \degrees + \cos 288 \degrees\)

\(\ds \)

\(=\)

\(\ds 0\)

Proof

Let $z_1, z_2, \dotsc, z_5$ be the complex $5$th roots of unity.

From Sum of Powers of Primitive Complex Roots of Unity, setting $s = 1$:

$z_1 + z_2 + z_3 + z_4 + z_5 = 0$

Thus from Complex 5th Roots of Unity:

$1 + \cis 72 \degrees + \cis 144 \degrees + \cis 216 \degrees + \cis 288 \degrees = 0$

Equating the real parts:

$1 + \cos 72 \degrees + \cos 144 \degrees + \cos 216 \degrees + \cos 288 \degrees = 0$

$\blacksquare$

Sources

• 1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: The $n$th Roots of Unity: $106$