Cosine of 3 Degrees

Theorem

$\cos 3 \degrees = \cos \dfrac {\pi} {60} = \dfrac {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2 + 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}$

where $\cos$ denotes the cosine function.

Proof

\(\ds \cos 3 \degrees\)

\(=\)

\(\ds \map \cos {75 \degrees - 72 \degrees}\)

\(\ds \)

\(=\)

\(\ds \cos 75 \degrees \cos 72 \degrees + \sin 75 \degrees \sin 72 \degrees\)

Cosine of Difference

\(\ds \)

\(=\)

\(\ds \dfrac {\sqrt 6 - \sqrt 2} 4 \times \dfrac {\sqrt 5 - 1} 4 + \dfrac {\sqrt 6 + \sqrt 2} 4 \times \dfrac {\sqrt {10 + 2 \sqrt5 } } 4\)

Cosine of $75 \degrees$, Cosine of $72 \degrees$, Sine of $75 \degrees$, Sine of $72 \degrees$

\(\ds \)

\(=\)

\(\ds \dfrac 1 {16} \times \paren {\paren {\sqrt 6 - \sqrt 2} \times \paren {\sqrt 5 - 1} + \paren {\sqrt 6 + \sqrt 2} \times \paren {\sqrt {10 + 2 \sqrt 5} } }\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {16} \times \paren {\paren {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2} + \paren {\sqrt {60 + 12 \sqrt 5} + \sqrt {20 + 4 \sqrt 5} } }\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {16} \times \paren {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2 + \sqrt {60 + 12 \sqrt 5} + \sqrt {20 + 4 \sqrt 5} }\)

\(\ds \)

\(=\)

\(\ds \dfrac 1 {16} \times \paren {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2 + 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} }\)

\(\ds \)

\(=\)

\(\ds \dfrac {\sqrt {30} - \sqrt {10} - \sqrt 6 + \sqrt 2 + 2 \sqrt {15 + 3 \sqrt 5} + 2 \sqrt {5 + \sqrt 5} } {16}\)

$\blacksquare$