Sine of 165 Degrees
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Theorem
- $\sin 165 \degrees = \sin \dfrac {11 \pi} {12} = \dfrac {\sqrt 6 - \sqrt 2} 4$
where $\sin$ denotes the sine function.
Proof
\(\ds \sin 165 \degrees\) | \(=\) | \(\ds \map \sin {90 \degrees + 75 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \cos 75 \degrees\) | Sine of Angle plus Right Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac {\sqrt 6 - \sqrt 2} 4\) | Cosine of $75 \degrees$ |
$\blacksquare$
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