Cosine of 30 Degrees

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Theorem

$\cos 30 \degrees = \cos \dfrac \pi 6 = \dfrac {\sqrt 3} 2$

where $\cos$ denotes the cosine.


Proof

\(\ds \paren {\cos 30 \degrees}^2\) \(=\) \(\ds 1 - \paren {\sin 30 \degrees}^2\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds 1 - \paren {\frac 1 2}^2\) Sine of $30 \degrees$
\(\ds \) \(=\) \(\ds \frac 3 4\)
\(\ds \leadsto \ \ \) \(\ds \cos 30 \degrees\) \(=\) \(\ds \sqrt {\frac 3 4}\)
\(\ds \) \(=\) \(\ds \dfrac {\sqrt 3} 2\) positive because $\cos 30 \degrees$ is in Quadrant I

$\blacksquare$


Sources

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