4 Sine Pi over 10 by Cosine Pi over 5/Proof 3
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Theorem
- $4 \sin \dfrac \pi {10} \cos \dfrac \pi 5 = 1$
Proof
\(\ds 4 \sin \theta \cos 2 \theta\) | \(=\) | \(\ds 1\) | Solve for $\theta$ | |||||||||||
\(\ds 4 \sin \theta \cos \theta \cos 2\theta\) | \(=\) | \(\ds \cos \theta\) | multiplying both sides by $\cos \theta$ | |||||||||||
\(\ds 2 \paren {2 \sin \theta \cos \theta } \cos 2\theta\) | \(=\) | \(\ds \cos \theta\) | factoring out $2$ | |||||||||||
\(\ds 2 \paren {\sin 2 \theta } \cos 2\theta\) | \(=\) | \(\ds \cos \theta\) | Double Angle Formula for Sine | |||||||||||
\(\ds \sin 4 \theta\) | \(=\) | \(\ds \cos \theta\) | Double Angle Formula for Sine | |||||||||||
\(\ds \map \sin {\frac \pi 2 - \theta}\) | \(=\) | \(\ds \cos \theta\) | Sine of Complement equals Cosine | |||||||||||
\(\ds \paren {\frac \pi 2 - \theta}\) | \(=\) | \(\ds 4 \theta\) | ||||||||||||
\(\ds \theta\) | \(=\) | \(\ds \frac \pi {10}\) |
$\blacksquare$