Difference between revisions of "Cosecant"

From specialfunctionswiki
Jump to: navigation, search
Line 1: Line 1:
 
The cosecant function is defined by
 
The cosecant function is defined by
$$\csc(z)=\dfrac{1}{\sin(z)}.$$
+
$$\csc(z)=\dfrac{1}{\sin(z)},$$
 +
where $\sin$ denotes the [[sine]] function.
  
 
<div align="center">
 
<div align="center">
Line 16: Line 17:
 
=See Also=
 
=See Also=
 
[[Arccsc]]<br />
 
[[Arccsc]]<br />
 +
[[Arccsch]] <br />
 
[[Csch]] <br />
 
[[Csch]] <br />
[[Arccsch]] <br />
+
[[Sine]]<b r/>
  
 
<center>{{:Trigonometric functions footer}}</center>
 
<center>{{:Trigonometric functions footer}}</center>

Revision as of 04:24, 8 February 2016

The cosecant function is defined by $$\csc(z)=\dfrac{1}{\sin(z)},$$ where $\sin$ denotes the sine function.

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
Arccsch
Csch
Sine<b r/>

<center>Trigonometric functions
</center>