Sine Rule (with proof)

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D

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A

B

a

C

b

none

Show Proof

$\mathsf{\text{In general}}$$\mathsf{\text{\textcolor{0,0,255}\frac{a}{sin\;A}={\textcolor{0,153,0}\frac{b}{sin\;B}={\textcolor{204,0,0}\frac{c}{sin\;C}}}}$

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\textcolor{102,0,102}\mathsf\text{Drag the \textcolor{red}{red} points at \textcolor{0,0,255}{A} and \textcolor{204,0,0}{C}.

From Δ ACD we get h = b sinA, and from Δ BCD we get h = a sinB.
Therefore b sinA = a sinB

Hence \;$\frac{b}{sin\;B}=\frac{a}{sin\;A}$

In \;general: $\frac{a}{sin\;A}=\frac{b}{sin\;B}=\frac{c}{sin\;C}

A = Point(xAxis)
C = (-6.164015536181704, 6.595233979881009)
B = Point(xAxis)
poly1, c, a, b = Polygon(A, B, C)
d = OrthogonalLine(C, c)
D = Intersect(d, xAxis)
e = Segment(C, D)
α = Angle(B, A, C)
β = Angle(C, B, A)
γ = Angle(A, C, B)
f = Segment(D, A)
text1: "A = " + α
text2: "B = " + β
text3: "C = " + γ
E = Midpoint(a)
F = Midpoint(c)
G = Midpoint(b)
text4: "a = " + a
text5: "c = " + c
text6: "b = " + b
δ = Angle(A, D, C)
H = Midpoint(e)
I = Midpoint(f)
g = false
J = Midpoint(D, B)
h = 1
M = (-14.865263982693309, 6.662685518226059)
N = (-12.066025141373684, 6.696411287398585)
i: a + b
j: sin(α)
k: a / j
text21: "$\mathsf{\textcolor{0,0,255}\frac{a}{sin\;A}={\textcolor{0,0,255}\frac{" + (LaTeX[a]) + "}{sin\;" + (LaTeX[α]) + "}=\textcolor{0,0,255}{" + (LaTeX[k]) + "$"
text12_1: "$\mathsf{\textcolor{0,153,0}\frac{b}{sin\;B}=\textcolor{0,153,0}\frac{" + (LaTeX[b]) + "}{sin\;" + (LaTeX[β]) + "}=\textcolor{0,153,0}{" + (LaTeX[k]) + "}$"
text12: "$\mathsf{\textcolor{204,0,0}\frac{c}{sin\;C}=\textcolor{204,0,0}\frac{" + (LaTeX[c]) + "}{sin\;" + (LaTeX[γ]) + "}=\textcolor{204,0,0}{" + (LaTeX[k]) + "}$"
K = (1.6895358426496978, 0.9237077258137203)
L = (1.6895358426496978, -0.49744288153366883)
l = Segment(K, L)
O = (6.770791765513562, 0.9574103287349301)
P = (6.770791765513562, -0.5412575844677708)
m = Segment(O, P)