Revision as of 23:37, 25 August 2015

The cosecant function is defined by $$\csc(z)=\dfrac{1}{\sin(z)}.$$

Properties

Theorem

The following formula holds: $$\dfrac{\mathrm{d}}{\mathrm{d}z} \csc(z)=- \cot(z)\csc(z),$$ where $\csc$ denotes the cosecant function and $\cot$ denotes the cotangent function.

Proof

Using the quotient rule and the definitions of cosecant and cotangent, $$\begin{array}{ll} \dfrac{\mathrm{d}}{\mathrm{d}z} \csc(z) &= \dfrac{\mathrm{d}}{\mathrm{d}z} \left[ \dfrac{1}{\sin(z)} \right] \\ &= \dfrac{0-\cos(z)}{\sin^2(z)} \\ &= -\csc(z)\cot(z), \end{array}$$ as was to be shown. █

References

• 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $4.3.108$

Theorem

The following formula holds: $$\csc(\mathrm{gd}(x))=\mathrm{coth}(x),$$ where $\csc$ is the cosecant, $\mathrm{gd}$ is the Gudermannian, and $\mathrm{coth}$ is the hyperbolic cotangent.

Proof

References

Theorem

The following formula holds: $$\mathrm{coth}(\mathrm{gd}^{-1}(x))=\csc(x),$$ where $\mathrm{coth}$ is the hyperbolic cotangent, $\mathrm{gd}^{-1}$ is the inverse Gudermannian, and $\csc$ is the cosecant.

Proof

References

Sine

Cosine

Tangent

Cosecant

Secant

Cotangent