Sine of 240 Degrees
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Theorem
- $\sin 240 \degrees = \sin \dfrac {4 \pi} 3 = -\dfrac {\sqrt 3} 2$
where $\sin$ denotes the sine function.
Proof 1
\(\ds \sin 240 \degrees\) | \(=\) | \(\ds \map \sin {360 \degrees - 120 \degrees}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds -\sin 120 \degrees\) | Sine of Conjugate Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds -\frac {\sqrt 3} 2\) | Sine of $120 \degrees$ |
$\blacksquare$
Proof 2
When $240 \degrees$ is embedded in a Cartesian plane, it makes an angle of $60 \degrees$ with the $x$-axis.
$240 \degrees$ can be found in the third quadrant.
Hence by definition of sine function in the third quadrant, $\sin 240 \degrees$ is negative.
Thus:
- $\sin 240 \degrees = -\sin 60 \degrees = -\dfrac {\sqrt 3} 2$
$\blacksquare$
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