Cosine of Conjugate Angle
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Theorem
- $\map \cos {2 \pi - \theta} = \cos \theta$
where $\cos$ denotes cosine.
That is, the cosine of an angle equals its conjugate.
Proof
| \(\ds \map \cos {2 \pi - \theta}\) | \(=\) | \(\ds \map \cos {2 \pi} \cos \theta + \map \sin {2 \pi} \sin \theta\) | Cosine of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1 \times \cos \theta + 0 \times \sin \theta\) | Cosine of Full Angle and Sine of Full Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds \cos \theta\) |
$\blacksquare$
Also see
- Sine 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