Sine of 75 Degrees
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Theorem
- $\sin 75 \degrees = \sin \dfrac {5 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$
where $\sin$ denotes the sine function.
Proof 1
\(\ds \sin 75 \degrees\) | \(=\) | \(\ds \map \sin {60 \degrees + 15 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sin 60 \degrees \cos 15 \degrees + \cos 60 \degrees \sin 15 \degrees\) | Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {\frac {\sqrt 3} 2} \paren {\frac {\sqrt 6 + \sqrt 2} 4} + \paren {\frac 1 2} \paren {\dfrac {\sqrt 6 - \sqrt 2} 4}\) | Sine of $60 \degrees$, Cosine of $15 \degrees$, Cosine of $60 \degrees$, Sine of $15 \degrees$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {\sqrt 3 \paren {\sqrt 6 + \sqrt 2} + \sqrt 6 - \sqrt 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {\sqrt 3 \sqrt 3 \sqrt 2 + \sqrt 3 \sqrt 2 + \sqrt 3 \sqrt 2 - \sqrt 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {3 \sqrt 2 + 2 \sqrt 3 \sqrt 2 - \sqrt 2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 8 \paren {2 \sqrt 2 + 2 \sqrt 6}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 + \sqrt 2} 4\) |
$\blacksquare$
Proof 2
\(\ds \sin 75 \degrees\) | \(=\) | \(\ds \map \cos {90 \degrees - 75 \degrees}\) | Cosine of Complement equals Sine | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos 15^\circ\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 + \sqrt 2} 4\) | Cosine of $15 \degrees$ |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Exact Values for Trigonometric Functions of Various Angles