edits
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jverzani
@@ -34,12 +34,37 @@ For a right triangle with angles $\theta$, $\pi/2 - \theta$, and $\pi/2$ ($0 < \
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```{julia}
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#| hide: true
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#| echo: false
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p = plot(legend=false, xlim=(-1/4,5), ylim=(-1/2, 3),
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xticks=nothing, yticks=nothing, border=:none)
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plot!([0,4,4,0],[0,0,3,0], linewidth=3)
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del = .25
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plot!([4-del, 4-del,4], [0, del, del], color=:black, linewidth=3)
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annotate!([(.75, .25, "θ"), (4.0, 1.25, "opposite"), (2, -.25, "adjacent"), (1.5, 1.25, "hypotenuse")])
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let
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gr()
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p = plot(;legend=false,
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xticks=nothing, yticks=nothing,
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border=:none,
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xlim=(-1/4,5), ylim=(-1/2, 3))
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plot!([0,4,4,0],[0,0,3,0], linewidth=3)
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θ = atand(3,4)
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del = .25
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plot!([4-del, 4-del,4], [0, del, del], color=:black, linewidth=3)
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theta = pi/20
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r = sqrt((3/4)^2 + (1/4)^2)
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ts = range(0, theta, 20)
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plot!(r*cos.(ts), r*sin.(ts); line=(:gray, 1))
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ts = range(atan(3/4) - theta, atan(3,4), 20)
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plot!(r*cos.(ts), r*sin.(ts); line=(:gray, 1))
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annotate!([
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(.75, .25, L"\theta"),
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(4.0, 1.5+.1, text("opposite", rotation=90,:top)),
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(2, -.25, "adjacent"),
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(2, 1.5+.1, text("hypotenuse", rotation=θ,:bottom))
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])
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end
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```
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```{julia}
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#| echo: false
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plotly()
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nothing
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```
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With these, the basic definitions for the primary trigonometric functions are
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@@ -77,9 +102,13 @@ gr()
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function plot_angle(m)
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r = m*pi
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n,d = numerator(m), denominator(m)
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ts = range(0, stop=2pi, length=100)
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tit = "$m ⋅ pi -> ($(round(cos(r), digits=2)), $(round(sin(r), digits=2)))"
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tit = latexstring("\\frac{$n}{$d}") *
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L"\cdot\pi\rightarrow (" *
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latexstring("$(round(cos(r), digits=2)),$(round(sin(r), digits=2)))")
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ts = range(0, 2pi, 151)
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p = plot(cos.(ts), sin.(ts), legend=false, aspect_ratio=:equal,title=tit)
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plot!(p, [-1,1], [0,0], color=:gray30)
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plot!(p, [0,0], [-1,1], color=:gray30)
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@@ -95,13 +124,11 @@ function plot_angle(m)
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plot!(p, [0,l*cos(r)], [0,l*sin(r)], color=:green, linewidth=4)
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scatter!(p, [cos(r)], [sin(r)], markersize=5)
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annotate!(p, [(1/4+cos(r), sin(r), "(x,y)")])
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annotate!(p, [(1/4+cos(r), sin(r), L"(x,y)")])
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p
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end
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## different linear graphs
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anim = @animate for m in -4//3:1//6:10//3
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plot_angle(m)
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@@ -406,7 +433,7 @@ Suppose both $\alpha$ and $\beta$ are positive with $\alpha + \beta \leq \pi/2$.
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```{julia}
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#| echo: false
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gr()
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using Plots, LaTeXStrings
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# two angles
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@@ -425,7 +452,10 @@ color1 = :royalblue
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color2 = :forestgreen
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color3 = :brown3
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color4 = :mediumorchid2
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canvas() = plot(axis=([],false), legend=false, aspect_ratio=:equal)
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canvas() = plot(;
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axis=([],false),
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legend=false,
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aspect_ratio=:equal)
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p1 = canvas()
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plot!(Shape([A,B,F]), fill=(color4, 0.15))
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@@ -440,29 +470,29 @@ ddf = sqrt(sum((D.-F).^2))
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Δ = 0.0
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alphabeta = (r*cos(α/2 + β/2), r*sin(α/2 + β/2),
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text("α + β",:hcenter; rotation=pi/2))
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cosαβ = (B[1]/2, 0, text("cos(α + β)", :top))
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sinαβ = (B[1], F[2]/2, text("sin(α + β)"))
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text(L"\alpha + \beta",:left, rotation=pi/2))
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cosαβ = (B[1]/2, 0, text(L"\cos(\alpha + \beta)", :top))
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sinαβ = (B[1], F[2]/2, text(L"\sin(\alpha + \beta)", rotation=90,:top))
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txtpoints = (
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one = (F[1]/2, F[2]/2, "1",:right),
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one = (F[1]/2, F[2]/2, text(L"1", :bottom)),
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beta=(r*cos(α + β/2), r*sin(α + β/2),
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text("β", :hcenter)),
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text(L"\beta", :hcenter)),
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alpha = (r*cos(α/2), r*sin(α/2),
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text("α",:hcenter)),
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text(L"\alpha",:hcenter)),
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alphaa = (F[1] + r*sin(α/2), F[2] - r*cos(α/2) ,
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text("α"),:hcenter),
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text(L"\alpha"),:hcenter),
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cosβ = (dae/2*cos(α),dae/2*sin(α) + Δ,
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text("cos(β)",:hcenter)),
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text(L"\cos(\beta)",:bottom, rotation=rad2deg(α))),
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sinβ = (B[1] + dbc/2 + Δ/2, D[2] + ddf/2 + Δ/2,
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text("sin(β)",:bottom)),
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cosαcosβ = (C[1]/2, 0 - Δ, text("cos(α)cos(β)", :top)),
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sinαcosβ = (cos(α)*cos(β) - 0.1, dce/2 ,
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text("sin(α)cos(β)", :hcenter)),
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text(L"\sin(\beta)",:bottom, rotation=-(90-rad2deg(α)))),
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cosαcosβ = (C[1]/2, 0 - Δ, text(L"\cos(\alpha)\cos(\beta)", :top)),
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sinαcosβ = (cos(α)*cos(β), dce/2 ,
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text(L"\sin(\alpha)\cos(\beta)", :top, rotation=90)),
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cosαsinβ = (D[1] - Δ, D[2] + ddf/2 ,
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text("cos(α)sin(β)", :top)),
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text(L"\cos(\alpha)\sin(\beta)", :bottom, 10, rotation=90)),
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sinαsinβ = (D[1] + dde/2, D[2] + Δ ,
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text("sin(α)sin(β)", :top)),
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text(L"\sin(\alpha)\sin(\beta)", 10, :top)),
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)
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# Plot 1
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@@ -574,6 +604,12 @@ $$
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$$
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:::
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```{julia}
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#| echo: false
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plotly()
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nothing
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```
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##### Example
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@@ -1028,4 +1064,3 @@ Is this identical to the pattern for the regular sine function?
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#| echo: false
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yesnoq(false)
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```
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