Teorema del seno

Teorema del coseno

Teorema del seno

Teorema del coseno

Teorema del seno

Teorema del coseno

Demostración

Comprobación

Sen\:\widehat{B}=\frac{h}{c}⇒h=c\:sen\widehat{B}

Sen\:\widehat{C}=\frac{h}{b}⇒h=b\:sen\widehat{C}

\frac{c}{sen\:\widehat{C}}=\frac{b}{sen\:\widehat{B}}

\frac{a}{sen\:\widehat{A}}=\frac{b}{sen\:\widehat{B}}=\frac{c}{sen\:\widehat{C}}

Demostración

Comprobación

c^2=h^2+x^2

x

a-x

b^2=(a-x)^2+h^2

b^2=a^2+x^2-2ax+h^2=a^2+c^2-2ax

cos\:\widehat{B}=\frac{x}{c}⇒x=c\:·cos\widehat{B}

b^2=a^2+c^2-2ac\:cos\widehat{B}⇒cos\widehat{B}=\frac{a^2+c^2-b^2}{2ac}

Razonando de igual forma sobre las otras dos alturas:

b^2=a^2+c^2-2\:a\:c\:cos\widehat{B}

a^2=b^2+c^2-2\:b\:c\:cos\widehat{A}

c^2=a^2+b^2-2\:a\:b\:cos\widehat{C}

a

b

c

c\:sen\widehat{B}=b\:sen\widehat{C}

h

th: 1
E1 = (-4, 1)
r_1 = Line(E1, xAxis)
E2 = Point(r_1)
E4 = Point(r_1)
E5 = Point(r_1)
E3 = Point(r_1)
s1 = Segment(E2, E3)
s2 = Segment(E4, E5)
B = Point(s1)
C = Point(s2)
r_3: x = x(B) + 0.5
r_2: x = x(C) - 0.5
r_4: y = y(B) + 0.5
r_5: y = y(B) + 5
F1 = Intersect(r_3, r_5)
F3 = Intersect(r_2, r_5)
F4 = Intersect(r_2, r_4)
F2 = Intersect(r_3, r_4)
poli1, f2, f1, f3, f4 = Polygon(F2, F1, F3, F4)
A = PointIn(poli1)
polígono1, c, b, a = Polygon(B, A, C)
d = OrthogonalLine(A, a)
H = Intersect(d, a)
h = Segment(A, H)
i1: false
i2: false
texto8: "b=" + b + ""
texto9: "c=" + c + ""
texto10: "a=" + a + ""
β = Angle(A, B, C)
γ = Angle(B, C, A)
α = Angle(B, A, C)
sena: sin(α)
senb: sin(β)
texto11: "a=" + (LaTeX[a]) + ";\:\:\:sen\:\widehat{A}=sen\:" + (LaTeX[α]) + "=" + (LaTeX[sena]) + ""
texto12: "b=" + (LaTeX[b]) + "\:;\:\:sen\:\widehat{B}=sen\:" + (LaTeX[β]) + "=" + (LaTeX[senb]) + ""
senc: sin(γ)
texto7: "c=" + (LaTeX[c]) + "\:;\:\:sen\:\widehat{C}=sen\:" + (LaTeX[γ]) + "=" + (LaTeX[senc]) + ""
csen: a / sena
texto13: "\frac{" + (LaTeX[a]) + "}{" + (LaTeX[sena]) + "}=\frac{" + (LaTeX[b]) + "}{" + (LaTeX[senb]) + "}=\frac{" + (LaTeX[c]) + "}{" + (LaTeX[senc]) + "}=" + (LaTeX[csen]) + ""
j1: false
j2: false
δ = Angle(A, H, B)
a2: a^(2)
b2: b^(2)
c2: c^(2)
cosa: cos(α)
cosb: cos(β)
cosc: cos(γ)
texto22: "cos\:\widehat{A}=" + (LaTeX[cosa]) + ""
texto24: "cos\:\widehat{B}=" + (LaTeX[cosb]) + ""
texto25: "cos\:\widehat{C}=" + (LaTeX[cosc]) + ""
texto27: "cos\widehat{B}=\frac{" + (LaTeX[a]) + "^2+" + (LaTeX[c]) + "^2-" + (LaTeX[b]) + "^2}{2·" + (LaTeX[a]) + "·" + (LaTeX[c]) + "}=" + (LaTeX[cosb]) + ""
texto28: "cos\widehat{A}=\frac{" + (LaTeX[b]) + "^2+" + (LaTeX[c]) + "^2-" + (LaTeX[a]) + "^2}{2·" + (LaTeX[b]) + "·" + (LaTeX[c]) + "}=" + (LaTeX[cosa]) + ""
texto29: "cos\widehat{C}=\frac{" + (LaTeX[a]) + "^2+" + (LaTeX[b]) + "^2-" + (LaTeX[c]) + "^2}{2·" + (LaTeX[a]) + "·" + (LaTeX[b]) + "}=" + (LaTeX[cosc]) + ""
texto32: "\widehat{B}=" + (LaTeX[β]) + ""
texto33: "\widehat{C}=" + (LaTeX[γ]) + ""
texto34: "\widehat{A}=" + (LaTeX[α]) + ""