Cosine of 15 Degrees/Proof 2
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Theorem
- $\cos 15 \degrees = \cos \dfrac \pi {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
Proof
\(\ds \cos 15 \degrees\) | \(=\) | \(\ds \map \cos {45 \degrees - 30 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos 45 \degrees \cos 30 \degrees + \sin 45 \degrees \sin 30 \degrees\) | Cosine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {\sqrt 2} 2} \paren {\frac {\sqrt 3} 2} + \paren {\frac {\sqrt 2} 2} \paren {\dfrac 1 2}\) | Cosine of $45 \degrees$, Cosine of $30 \degrees$, Sine of $45 \degrees$, Sine of $30 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 6} 4 + \frac {\sqrt 2} 4\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 + \sqrt 2} 4\) |
$\blacksquare$