naginterfaces.library.specfun.tan

naginterfaces.library.specfun.tan(x)

tan returns the value of the circular tangent, $\tan \left(x\right)$.

For full information please refer to the NAG Library document for s07aa

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/s/s07aaf.html

Parameters

xfloat

The argument $x$ of the function.

Returns

txfloat

The value of the circular tangent, $\tan \left(x\right)$.

Raises

NagValueError

(errno $1$)

On entry, $x=⟨value⟩$ and the constant $F_1=⟨value⟩$.

Constraint: $\left|x\right|\le F_1$.

Warns

NagAlgorithmicWarning

(errno $2$)

On entry, $x=⟨value⟩$ and the constant $F_2=⟨value⟩$.

The function has been called with an argument that is too close to an odd multiple of $\pi /2$, at which the function is infinite; the function has returned a value with the correct sign but a more or less arbitrary but large magnitude.

Notes

No equivalent traditional C interface for this routine exists in the NAG Library.

tan calculates an approximate value for the circular tangent of its argument, $\tan \left(x\right)$. It is based on the Chebyshev expansion

$$\tan \left(\theta \right)=\theta y\left(t\right)=\theta {\sum ^{\prime }}_{r=0}{c}_r{T}_r\left(t\right)$$

where $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$ and $-1<t<+1\text{,}t=2{\left(\frac{4\theta }{\pi }\right)}^2-1$.

The reduction to the standard range is accomplished by taking

$$x=N\pi /2+\theta$$

where $N$ is an integer and $-\frac{\pi }{4}<\theta <\frac{\pi }{4}$,

i.e., $\theta =x-\left(\frac{2x}{\pi }\right)\frac{\pi }{2}$ where $N=\left[\frac{2x}{\pi }\right]=\text{the nearest integer to}\frac{2x}{\pi }$.

From the properties of $\tan \left(x\right)$ it follows that

$$\begin{array}{c}\tan \left(x\right)=\left\{\begin{array}{cc}\tan \left(\theta \right)\text{,}& Neven\\ -1/\tan \left(\theta \right)\text{,}& Nodd\end{array}\right\}\end{array}$$

References

NIST Digital Library of Mathematical Functions