Sum of Cosines of k pi over 5

Theorem

$\cos 36 \degrees + \cos 72 \degrees + \cos 108 \degrees + \cos 144 \degrees = 0$

Proof

We have:

\(\ds 144 \degrees\)

\(=\)

\(\ds 180 \degrees - 36 \degrees\)

\(\ds \leadsto \ \ \)

\(\ds \cos 36 \degrees\)

\(=\)

\(\ds -\cos 144 \degrees\)

Cosine of Supplementary Angle

and:

\(\ds 108 \degrees\)

\(=\)

\(\ds 180 \degrees - 72 \degrees\)

\(\ds \leadsto \ \ \)

\(\ds \cos 72 \degrees\)

\(=\)

\(\ds -\cos 108 \degrees\)

Cosine of Supplementary Angle

Thus:

$\cos 36 \degrees + \cos 72 \degrees + \cos 108 \degrees + \cos 144 \degrees = 0$

$\blacksquare$

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: $107$