Sine of Conjugate Angle
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Theorem
- $\map \sin {2 \pi - \theta} = -\sin \theta$
where $\sin$ denotes sine.
That is, the sine of an angle is the negative of its conjugate.
Proof
\(\ds \map \sin {2 \pi - \theta}\) | \(=\) | \(\ds \map \sin {2 \pi} \cos \theta - \map \cos {2 \pi} \sin \theta\) | Sine of Difference | |||||||||||
\(\ds \) | \(=\) | \(\ds 0 \times \cos \theta - 1 \times \sin \theta\) | Sine of Full Angle and Cosine of Full Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds -\sin \theta\) |
$\blacksquare$
Also see
- Cosine of Conjugate Angle
- Tangent of Conjugate Angle
- Cotangent of Conjugate Angle
- Secant of Conjugate Angle
- Cosecant of Conjugate Angle
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I