Trigonometric Functions of Complementary Angles

Theorem

Sine of Complement equals Cosine

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

Cosine of Complement equals Sine

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

Tangent of Complement equals Cotangent

$\map \tan {\dfrac \pi 2 - \theta} = \cot \theta$ for $\theta \ne n \pi$

where $\tan$ and $\cot$ are tangent and cotangent respectively.

That is, the cotangent of an angle is the tangent of its complement.

This relation is defined wherever $\sin \theta \ne 0$.

Cotangent of Complement equals Tangent

Let $\theta \ne \paren {2 n + 1} \dfrac \pi 2$

Then:

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

Secant of Complement equals Cosecant

$\map \sec {\dfrac \pi 2 - \theta} = \csc \theta$ for $\theta \ne n \pi$

where $\sec$ and $\csc$ are secant and cosecant respectively.

That is, the cosecant of an angle is the secant of its complement.

This relation is defined wherever $\sin \theta \ne 0$.

Cosecant of Complement equals Secant

$\map \csc {\dfrac \pi 2 - \theta} = \sec \theta$ for $\theta \ne \paren {2 n + 1} \dfrac \pi 2$

where $\csc$ and $\sec$ are cosecant and secant respectively.

That is, the secant of an angle is the cosecant of its complement.

This relation is defined wherever $\cos \theta \ne 0$.

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