use quarto, not Pluto to render pages
This commit is contained in:
jverzani
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# Trigonometric functions
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This section uses the following add-on packages:
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```julia
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@@ -11,6 +12,8 @@ using SymPy
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```julia; echo=false; results="hidden"
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using CalculusWithJulia.WeaveSupport
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fig_size = (800, 600)
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const frontmatter = (
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title = "Trigonometric functions",
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description = "Calculus with Julia: Trigonometric functions",
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@@ -66,9 +69,8 @@ trigonometric functions are
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\end{align*}
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```
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```julia; echo=false
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note("""Many students remember these through [SOH-CAH-TOA](http://mathworld.wolfram.com/SOHCAHTOA.html).""")
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```
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!!! note
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Many students remember these through [SOH-CAH-TOA](http://mathworld.wolfram.com/SOHCAHTOA.html).
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Some algebra shows that $\tan(\theta) = \sin(\theta)/\cos(\theta)$. There are also ``3`` reciprocal functions, the cosecant, secant and cotangent.
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@@ -78,7 +80,6 @@ These definitions in terms of sides only apply for $0 \leq \theta \leq \pi/2$. M
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```julia; hold=true; echo=false; cache=true
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## {{{radian_to_trig}}}
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fig_size = (400, 300)
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function plot_angle(m)
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r = m*pi
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@@ -181,15 +182,10 @@ sincos(pi/3)
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```
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```julia; echo=false
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note(L"""
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!!! note
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For really large values, round off error can play a big role. For example, the *exact* value of $\sin(1000000 \pi)$ is $0$, but the returned value is not quite $0$ `sin(1_000_000 * pi) = -2.231912181360871e-10`. For exact multiples of $\pi$ with large multiples the `sinpi` and `cospi` functions are useful.
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For really large values, round off error can play a big role. For example, the *exact* value of $\sin(1000000 \pi)$ is $0$, but the returned value is not quite $0$ `sin(1_000_000 * pi) = -2.231912181360871e-10`. For exact multiples of $\pi$ with large multiples the `sinpi` and `cospi` functions are useful.
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(Both functions are computed by first employing periodicity to reduce the problem to a smaller angle. However, for large multiples the floating-point roundoff becomes a problem with the usual functions.)
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""")
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```
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(Both functions are computed by first employing periodicity to reduce the problem to a smaller angle. However, for large multiples the floating-point roundoff becomes a problem with the usual functions.)
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##### Example
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@@ -356,7 +352,6 @@ end
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# create animoation
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b₁=1/3; n₁=3; b₂=1/4; n₂=4
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fig_size = (400, 300)
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anim = @animate for t ∈ range(0, 2.5, length=50)
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makegraph(t, b₁, n₁, b₂, n₂)
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end
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@@ -618,45 +613,47 @@ the unit *hyperbola* ($x^2 - y^2 = 1$). We define the hyperbolic
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sine ($\sinh$) and hyperbolic cosine ($\cosh$) through $(\cosh(\theta),
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\sinh(\theta)) = (x,y)$.
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```julia; hold=true; echo=false
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## inspired by https://en.wikipedia.org/wiki/Hyperbolic_function
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# y^2 = x^2 - 1
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top(x) = sqrt(x^2 - 1)
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```julia; echo=false
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let
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## inspired by https://en.wikipedia.org/wiki/Hyperbolic_function
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# y^2 = x^2 - 1
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top(x) = sqrt(x^2 - 1)
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p = plot(; legend=false, aspect_ratio=:equal)
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p = plot(; legend=false, aspect_ratio=:equal)
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x₀ = 2
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xs = range(1, x₀, length=100)
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ys = top.(xs)
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plot!(p, xs, ys, color=:red)
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plot!(p, xs, -ys, color=:red)
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x₀ = 2
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xs = range(1, x₀, length=100)
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ys = top.(xs)
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plot!(p, xs, ys, color=:red)
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plot!(p, xs, -ys, color=:red)
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xs = -reverse(xs)
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ys = top.(xs)
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plot!(p, xs, ys, color=:red)
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plot!(p, xs, -ys, color=:red)
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xs = -reverse(xs)
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ys = top.(xs)
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plot!(p, xs, ys, color=:red)
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plot!(p, xs, -ys, color=:red)
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xs = range(-x₀, x₀, length=3)
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plot!(p, xs, xs, linestyle=:dash, color=:blue)
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plot!(p, xs, -xs, linestyle=:dash, color=:blue)
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xs = range(-x₀, x₀, length=3)
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plot!(p, xs, xs, linestyle=:dash, color=:blue)
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plot!(p, xs, -xs, linestyle=:dash, color=:blue)
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a = 1.2
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plot!(p, [0,cosh(a)], [sinh(a), sinh(a)])
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annotate!(p, sinh(a)/2, sinh(a)+0.25,"cosh(a)")
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plot!(p, [cosh(a),cosh(a)], [sinh(a), 0])
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annotate!(p, sinh(a) + 1, cosh(a)/2,"sinh(a)")
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scatter!(p, [cosh(a)], [sinh(a)], markersize=5)
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a = 1.2
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plot!(p, [0,cosh(a)], [sinh(a), sinh(a)])
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annotate!(p, [(sinh(a)/2, sinh(a)+0.25,"cosh(a)")])
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plot!(p, [cosh(a),cosh(a)], [sinh(a), 0])
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annotate!(p, [(sinh(a) + 1, cosh(a)/2,"sinh(a)")])
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scatter!(p, [cosh(a)], [sinh(a)], markersize=5)
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ts = range(0, a, length=100)
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xs′ = cosh.(ts)
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ys′ = sinh.(ts)
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ts = range(0, a, length=100)
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xs′ = cosh.(ts)
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ys′ = sinh.(ts)
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xs = [0, 1, xs′..., 0]
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ys = [0, 0, ys′..., 0]
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plot!(p, xs, ys, fillcolor=:red, fill=true, alpha=.3)
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xs = [0, 1, xs′..., 0]
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ys = [0, 0, ys′..., 0]
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plot!(p, xs, ys, fillcolor=:red, fill=true, alpha=.3)
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p
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p
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end
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```
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These values are more commonly expressed using the exponential function as:
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@@ -682,8 +679,8 @@ What is bigger $\sin(1.23456)$ or $\cos(6.54321)$?
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```julia; hold=true; echo=false
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a = sin(1.23456) > cos(6.54321)
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choices = [raw"``\sin(1.23456)``", raw"``\cos(6.54321)``"]
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ans = a ? 1 : 2
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radioq(choices, ans, keep_order=true)
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answ = a ? 1 : 2
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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@@ -694,8 +691,8 @@ Let $x=\pi/4$. What is bigger $\cos(x)$ or $x$?
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x = pi/4
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a = cos(x) > x
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choices = [raw"``\cos(x)``", "``x``"]
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ans = a ? 1 : 2
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radioq(choices, ans, keep_order=true)
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answ = a ? 1 : 2
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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@@ -707,8 +704,8 @@ choices = [
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raw"``\cos(x) = \sin(x - \pi/2)``",
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raw"``\cos(x) = \sin(x + \pi/2)``",
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raw"``\cos(x) = \pi/2 \cdot \sin(x)``"]
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ans = 2
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radioq(choices, ans)
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answ = 2
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radioq(choices, answ)
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```
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###### Question
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@@ -720,8 +717,8 @@ choices = [
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L"The values $k\pi$ for $k$ in $\dots, -2, -1, 0, 1, 2, \dots$",
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L"The values $\pi/2 + k\pi$ for $k$ in $\dots, -2, -1, 0, 1, 2, \dots$",
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L"The values $2k\pi$ for $k$ in $\dots, -2, -1, 0, 1, 2, \dots$"]
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ans = 2
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radioq(choices, ans, keep_order=true)
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answ = 2
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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@@ -768,24 +765,24 @@ The sine function is an *odd* function.
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```julia; hold=true; echo=false
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choices = ["odd", "even", "neither"]
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ans = 1
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radioq(choices, ans, keep_order=true)
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answ = 1
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radioq(choices, answ, keep_order=true)
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```
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* The hyperbolic cosine is:
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```julia; hold=true; echo=false
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choices = ["odd", "even", "neither"]
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ans = 2
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radioq(choices, ans, keep_order=true)
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answ = 2
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radioq(choices, answ, keep_order=true)
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```
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* The hyperbolic tangent is:
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```julia; hold=true; echo=false
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choices = ["odd", "even", "neither"]
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ans = 1
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radioq(choices, ans, keep_order=true)
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answ = 1
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radioq(choices, answ, keep_order=true)
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```
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###### Question
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