Difference between revisions of "Cosecant"

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File:Cosecantplot.png|Graph of $\csc$ on $[-2\pi,2\pi]$.
 
File:Cosecantplot.png|Graph of $\csc$ on $[-2\pi,2\pi]$.
File:Complex Csc.jpg|[[Domain coloring]] of [[analytic continuation]] of $\csc$.
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File:Complexcosecantplot.png|[[Domain coloring]] of $\csc$.
 
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Revision as of 04:24, 8 February 2016

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

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

See Also

Arccsc
Csch
Arccsch

<center>Trigonometric functions
</center>