Trigonometric Functions of Conjugate Angles

Theorem

Sine of Conjugate Angle

$\map \sin {2 \pi - \theta} = -\sin \theta$

where $\sin$ denotes sine.

That is, the sine of an angle is the negative of its conjugate.

Cosine of Conjugate Angle

$\map \cos {2 \pi - \theta} = \cos \theta$

where $\cos$ denotes cosine.

That is, the cosine of an angle equals its conjugate.

Tangent of Conjugate Angle

$\map \tan {2 \pi - \theta} = -\tan \theta$

where $\tan$ denotes tangent.

That is, the tangent of an angle is the negative of its conjugate.

Cotangent of Conjugate Angle

$\map \cot {2 \pi - \theta} = -\cot \theta$

where $\cot$ denotes cotangent.

That is, the cotangent of an angle is the negative of its conjugate.

Secant of Conjugate Angle

$\map \sec {2 \pi - \theta} = \sec \theta$

where $\sec$ denotes secant.

That is, the secant of an angle equals its conjugate.

Cosecant of Conjugate Angle

$\map \csc {2 \pi - \theta} = -\csc \theta$

where $\csc$ denotes cosecant.

That is, the cosecant of an angle is the negative of its conjugate.

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