functions tan and cot with multiples of pi

[raneameya]

Is it reasonable to expect tan(π/2) == Inf to be true? Similarly, should tan(π) == 0 be true?

Also, not to labour the point, but I noticed that cot(π) threw a MethodError.

julia> versioninfo()
Julia Version 1.0.0
Commit 5d4eaca0c9 (2018-08-08 20:58 UTC)
Platform Info:
  OS: Windows (x86_64-w64-mingw32)
  CPU: Intel(R) Core(TM) i7-7700 CPU @ 3.60GHz
  WORD_SIZE: 64
  LIBM: libopenlibm
  LLVM: libLLVM-6.0.0 (ORCJIT, skylake)
Environment:
  JULIA_EDITOR = "C:\XXXX\atom\app-1.29.0\atom.exe" -a
  JULIA_NUM_THREADS = 4

[pochoi]

On the other hand, the functions sinpi and cospi work as you would expect.

julia> sinpi(0.5)/cospi(0.5) == Inf
true
julia> sinpi(1.0)/cospi(1.0) == 0.0
true

Unfortunately there is no tanpi yet.

---

You made a good point. cot(π) will call one(Irrational{:π}) / tan(π). However, one(Irrational{:π}) is not defined. Same situation for csc and sec. It seems that the conversion of type Irrational is not perfect at this moment.

[stevengj]

See also #7994 and #5561.

[ravibitsgoa]

Can we define a tanpi(x) method as sinpi(x)/cospi(x)?
And similarly for cotpi, cscpi, secpi?

[gekaremi]

Is it good practice to replace tangent and cotangent with sine-cosine ratios?

I stumbled on the fact that in Julia there is no this function (cotpi and tanpi), and in numpy (as another example) there is no cotangent at all, so I would like to know what can be done?

(sorry for bad English)

[StefanKarpinski]

Defining accurate versions of all of these trig functions for multiples of pi seems like the right way.

[laborg]

at least tanpi has been added in 1.10

[waldyrious]

Why was this issue closed, @oscardssmith? While there's tanpi now, there still isn't a cotpi, cscpi, or secpi.

[oscardssmith]

fair enough

[11happy]

I am interested in the issue and have understood the full thread.
Here's my proposed solution :
cotpi:

function cotpi(x::T) where T<:IEEEFloat
    tanpi(x) == 0 && throw(DivideError("cotpi: cotangent is undefined for argument $x"))
    return 1.0 / tanpi(x)
end

secpi:

function secpi(x::T) where T<:IEEEFloat
    cospi(x) == 0 && throw(DivideError("secpi: secant is undefined for argument $x"))
    return 1.0 / cospi(x)
end

cscpi:

function cscpi(x::T) where T<:IEEEFloat
    sinpi(x) == 0 && throw(DivideError("cscpi: cosecant is undefined for argument $xr"))
    return 1.0 / sinpi(x)
end

If someone could please review these and provide any feedback or suggestions, it would be very helpful .
Thank you

[Youjack]

Current definition of tanpi produces NaN for large complex argument. For example,

julia> tanpi(500 * exp(im * 3pi/4))
NaN + NaN*im

The reason is that sinpi and cospi produce Inf with this argument,

julia> sinpi(500 * exp(im * 3pi/4))
Inf + Inf*im

julia> cospi(500 * exp(im * 3pi/4))
Inf - Inf*im

while tanpi is defined as

julia/base/special/trig.jl

Line 943 in 3318941

tanpi(x::Complex) = sinpi(x) / cospi(x) # Is there a better way to do this?

This NaN causes NaN in some special functions. For example,

julia> using SpecialFunctions

julia> digamma(500 * exp(im * 3pi/4))
NaN + NaN*im

See the related issue in SpecialFunctions.jl repo for details.

A possible way to resolve this problem is to define tanpi as how cotpi is defined in mpmath:
https://github.com/mpmath/mpmath/blob/b600dbcabf4b7406a61e82b9e607f754a9f12ff9/mpmath/libfp.py#L153-L161