Sine of 105 Degrees

Theorem

$\sin 105^\circ = \sin \dfrac {7 \pi} {12} = \dfrac {\sqrt 6 + \sqrt 2} 4$

where $\sin$ denotes the sine function.


Proof

\(\ds \sin 105^\circ\) \(=\) \(\ds \sin \left({90^\circ + 15^\circ}\right)\)
\(\ds \) \(=\) \(\ds \cos 15^\circ\) Sine of Angle plus Right Angle
\(\ds \) \(=\) \(\ds \frac {\sqrt 6 + \sqrt 2} 4\) 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
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