I wonder why the sin(x) function can’t be plot for the domain of -π or the negative x-axis.

This is my code:

using Plots, LaTeXStrings
gr()
function pitick(start, stop, denom; mode=:text)
a = Int(cld(start, π/denom))
b = Int(fld(stop, π/denom))
tick = range(a*π/denom, b*π/denom; step=π/denom)
ticklabel = piticklabel.((a:b) .// denom, Val(mode))
tick, ticklabel
end
function piticklabel(x::Rational, ::Val{:text})
iszero(x) && return "0"
S = x < 0 ? "-" : ""
n, d = abs(numerator(x)), denominator(x)
N = n == 1 ? "" : repr(n)
d == 1 && return S * N * "π"
S * N * "π/" * repr(d)
end
function piticklabel(x::Rational, ::Val{:latex})
iszero(x) && return L"0"
S = x < 0 ? "-" : ""
n, d = abs(numerator(x)), denominator(x)
N = n == 1 ? "" : repr(n)
d == 1 && return L"%$S%$N\pi"
L"%$S\frac{%$N\pi}{%$d}"
end
a, b = -π, 2π/3
f(x) = (sin.(x))^(2/3)
plot(f, a, b; xtick=pitick(a, b, 4; mode=:latex), framestyle=:zerolines,
legend=:outerright, label=L"f(x) = (\sin \ x)^{2/3}",
size=(720, 360), tickfontsize=10)

You can’t raise a negative real number to a non-integer power:

julia> sin(-1.)
-0.8414709848078965
julia> ans^(2/3)
ERROR: DomainError with -0.8414709848078965:
Exponentiation yielding a complex result requires a complex argument.
Replace x^y with (x+0im)^y, Complex(x)^y, or similar.

I think you need something like f(x) = real((complex(sin.(x)))^(2/3)) which makes a plot. I’m not sure if it’s the plot you want, but it plots over the negative part of the x axis.

Just use f(x) = cbrt(sin(x))^2 to compute the 2/3 power in a way that works for negative numbers (and is faster to boot). (Not sure why you have a dot here. If you want to apply f elementwise to a vector just do f.(x).)

That’s what the cbrt function is for, whereas converting to complex chooses a different branch cut that you may not want here:

# return the real x^p (if it exists)
realpow(x::Real, p::Rational) =
isodd(p.den) || x ≥ 0 ? copysign(abs(x)^(1//p.den), x)^p.num : throw(DomainError())

and then call realpow(sin(x), 2//5).

(In general, most math libraries in any language tend not to provide this kind of function, as far as I can tell, because it is a nonstandard branch cut for the x^(1/n) power.)

Yes, if that’s what you mean by x^{7/5} (for negative x, the real 5th root, computed by nthroot(x, 5) is different from the principal nth root, computed by complex(x)^(1//5)).