Add trigonometric functions for special irrationals

[devmotion]

Fixes #20.

This PR is a (largely) reduced and updated version of #14. Arguably it is quite uncontroversial since it just matches the existing behaviour of sin(pi) etc. in Base: JuliaLang/julia#42595

[codecov]

Codecov Report

Base: 97.43% // Head: 98.24% // Increases project coverage by +0.80% 🎉

Coverage data is based on head (dd8b355) compared to base (6ecf42b).
Patch coverage: 100.00% of modified lines in pull request are covered.

Additional details and impacted files

@@            Coverage Diff             @@
##             main      #21      +/-   ##
==========================================
+ Coverage   97.43%   98.24%   +0.80%     
==========================================
  Files           2        3       +1     
  Lines          39       57      +18     
==========================================
+ Hits           38       56      +18     
  Misses          1        1              

Impacted Files	Coverage Δ
src/IrrationalConstants.jl	100.00% <ø> (ø)
src/trigonometric.jl	100.00% <100.00%> (ø)

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[coveralls]

Pull Request Test Coverage Report for Build 4254120663

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Details

• 16 of 16 (100.0%) changed or added relevant lines in 1 file are covered.
• No unchanged relevant lines lost coverage.
• Overall coverage remained the same at 100.0%

---

Totals
Change from base Build 4223760967:	0.0%
Covered Lines:	53
Relevant Lines:	53

---

💛 - Coveralls

[hyrodium]

Note that sec(quartπ) does not have consistent accuracy because sec(x) is defined by inv(cos(x)).

julia> sec(quartπ)
1.414213562373095

julia> Float64(sqrt2)
1.4142135623730951

julia> inv(invsqrt2)  # same as inv(cos(invsqrt2))
1.414213562373095

This is one of the reasons why I would like to have sec(quartπ) === sqrt2 in #14.

[devmotion]

This is one of the reasons why I would like to have sec(quartπ) === sqrt2 in #14.

I think we have to be consistent and always return Float64 (consistent with Base), otherwise it gets completely confusing. I improved the accuracy slightly by defining sec and csc for quartπ.

[hyrodium]

I think we have to be consistent and always return Float64 (consistent with Base),

There is a counter example log(ℯ) === 1 (#14 (comment)).

otherwise it gets completely confusing.

I thought "return Irrational if we have the irrational, otherwise, return Float64" is the best approach (#14 (comment)), and I could not find that much confusion here.

However, I agree with merging this PR because this PR just improves the accuracy and there are no breaking changes. My last question is why my PR #14 is not merged yet. I updated the PR to return Float64 for all Irrational inputs as you suggested.

[devmotion]

There is a counter example

The log definition is much older than the trigonometric definitions in Base and the odd one, it seems. I think when defining trigonometric functions for irrationals we should be consistent with how in Base trigonometric funtions for irrationals are implemented. And the reasoning there was quite clear, they should all return Float64.

My last question is why my PR #14 is not merged yet. I updated the PR to return Float64 for all Irrational inputs as you suggested.

It seems unfortunately I lost track of it, and with the recent changes it became outdated. Hence I decided to open a new PR. Generally, I also prefer smaller PRs with fewer changes (e.g., focused only on trigonometric functions) and without additional formatting changes.