Formules de trigonométrie
Author: Yvan Monka - m@ths et tiques 2013

-1

-\frac{\sqrt{3}}{2}

-\frac{\sqrt{2}}{2}

-\frac{1}{2}

0

\frac{1}{2}

\frac{\sqrt{2}}{2}

\frac{\sqrt{3}}{2}

1

0

Déplacer M

π-x

π+x

π/2-x

1

1

-x

π/2+x

$ cos\left(  -x \right)=cos\left(  x \right) \; \; sin\left(  -x \right)=-sin\left(  x \right)  $

$ cos\left(  \pi +x \right)=-cos\left(  x \right) \; \; sin\left(  \pi +x \right)=-sin\left(  x \right)  $

$ cos\left(  \pi -x \right)=-cos\left(  x \right) \; \; sin\left(  \pi -x \right)=sin\left(  x \right)  $

$ cos\left(  \frac{\pi}{2} -x \right)=sin\left(  x \right) \; \; sin\left(  \frac{\pi}{2}-x \right)=cos\left(  x \right)  $

$ cos\left(  \frac{\pi}{2} +x \right)=-sin\left(  x \right) \; \; sin\left(  \frac{\pi}{2}+x \right)=cos\left(  x \right)  $

l1: {"   0  ", "\frac{ π}{6}", "\frac{ π}{4}", "\frac{π}{3}", "\frac{π}{2}", "\frac{2π}{3}", "\frac{3π}{4}", "\frac{5π}{6}", "  π"}
B1 = 2
B2 = 2
B3 = 2
B4 = 2
B5 = 2
B6 = 2
B7 = 2
B8 = 2
B9 = 2
s1 = Join({Sequence[LaTeX[Object["A" + (i)]], i, 5, 9], Sequence[LaTeX[Object["A" + (i)]], i, 8, 5, -1]})
l = Join({Sequence["-" + (Element[l1, i]), i, 8, 2, -1], l1})
c1 = Sequence(LaTeX[Object["A" + (i)]], i, 9, 1, -1)
s = Join({Sequence["-" + (Element[s1, i]), i, 8, 2, -1], s1})
c_2 = Join({Sequence[Element[c1, i], i, 8, 2, -1], c1})
l2 = Append("x", l)
c2 = Append("cos(x)", c_2)
s2 = Append("sin(x)", s)
A = Intersect(xAxis, yAxis)
c = Circle(A, 1)
v: {-2.6179938779914944, -2.356194490192345, -2.0943951023931953, -1.5707963267948966, -1.0471975511965976, -0.7853981633974483, -0.5235987755982988, 0, 0.5235987755982988, 0.7853981633974483, 1.0471975511965976, 1.5707963267948966, 2.0943951023931953, 2.356194490192345, 2.6179938779914944, 3.141592653589793}
a = 3 (slider from -7 to 8, step 1)
M: (cos(Element[v, a + 8]), sin(Element[v, a + 8]))
cos: (x(M), 0)
sin: (0, y(M))
b = Segment(cos, M)
e = Segment(M, sin)
g = false
h = false
i = Segment(M, A)
M' = Mirror(M, xAxis)
j = Segment(A, M')
M'_1 = Mirror(M, yAxis)
k = Segment(M'_1, A)
M'_2 = Mirror(M, A)
m = Segment(A, M'_2)
n = false
p = Segment(cos, M')
B: (0, -1 * y(M))
C: (-1 * x(M), 0)
q = Segment(B, M')
r = Segment(M'_1, sin)
t = Segment(M'_1, C)
a_1 = Segment(C, M'_2)
b_1 = Segment(M'_2, B)
M'_3 = Mirror(M, c_1)
d_1 = Segment(M'_3, A)
D: (0, x(M))
E: (y(M), 0)
e_1 = Segment(M'_3, E)
f_1 = Segment(M'_3, D)
i_1 = Segment((-0.8, -0.8), (0.8, 0.8))
valpart = LaTeX(TableText[{l2, c2, s2}, "lc"])
xg: x(Corner[valpart, 1])
yb: y(Corner[valpart, 1])
yh: y(Corner[valpart, 3])
xd: x(Corner[valpart, 3])
k10: {0, 0.03, 0, -0.03}
f10: {0, 0.23, 0.03, 0.02, 0.01, 0, 0, 0.01, 0, -0.1, -0.04, -0.02, -0.07, -0.07, -0.02, 0, 0.03, -0.02}
k1 = Sequence(Segment[(xg, yb + Sum[First[k10, i + 1]] + i * (yh - yb) / 3), (xd, yb + Sum[First[k10, i + 1]] + i * (yh - yb) / 3)], i, 0, 3)
f = Sequence(Segment[(xg + Sum[First[f10, i + 1]] + i * (xd - xg) / 17, yb), (xg + i * (xd - xg) / 17 + Sum[First[f10, i + 1]], yh)], i, 0, 17)
texte5: "  " + (LaTeX[Element[l, a + 8]]) + ""
F = (0, 1)
u = Vector(A, F)
G = (1, 0)
w = Vector(A, G)
α = Angle(G, A, M)
H = (0.8508742099723046, 0.003145008981470277)
I = Midpoint(i)
J = Midpoint(I, H)
β = Angle(M', A, G)
f_2 = true
l1_1: {"   0  ", "\frac{ π}{6}", "\frac{ π}{4}", "\frac{π}{3}", "\frac{π}{2}", "\frac{2π}{3}", "\frac{3π}{4}", "\frac{5π}{6}", "  π"}
l_1 = Join({Sequence["-" + (Element[l1_1, i]), i, 8, 2, -1], l1_1})
texte5_1: "x =  " + (LaTeX[Element[l_1, a + 8]]) + ""
γ = Angle(G, A, M'_2)
δ = Angle(G, A, M'_1)
ε = Angle(G, A, M'_3)
A' = Rotate(M, 90*°, A)
g_1 = Segment(A, A')
h_1 = OrthogonalLine(A', yAxis)
j_1 = OrthogonalLine(A', xAxis)
K = Intersect(j_1, xAxis)
L = Intersect(h_1, yAxis)
k_1 = Segment(L, A')
m_1 = Segment(A', K)
o = false
ζ = Angle(G, A, A')
"\bgcolor{30ee00}{cos(" + (LaTeX[Elément[l, a + 8]]) + ") = " + (LaTeX[Elément[c, 8 + a]]) + "" + (LaTeX[Si[f ∧ (a < 8), "\fgcolor{ff0000}{=cos(" + (Elément[l, -a + 8]) + ")}", ""]]) + "" + (LaTeX[Si[g, "\fgcolor{ff00ff}{=-cos(" + (Si[a < 0, "-", ""]) + (Elément[l, -abs(a) + 16]) + ")} ", ""]]) + "" + (LaTeX[Si[h, "\fgcolor{0000ff}{=-cos(" + (Si[a ≤ 0, Elément[l, a + 16], Elément[l, a]]) + ")}", ""]]) + "" + (LaTeX[Si[n, "=sin(" + (Si[a < -1 (4), Elément[l, -4 - a], Elément[l, 12 - a]]) + ")", ""]]) + "}": "\fgcolor{0000ff}{cos \left(    " + (LaTeX[Element[l, a + 8]]) + "\right) = " + (LaTeX[Element[c_2, 8 + a]]) + "}" + (LaTeX[If[f_2 ∧ (a < 8), "\fgcolor{ff00ff}{=cos\left(" + (Element[l, -a + 8]) + "\right)}", ""]]) + "" + (LaTeX[If[g, "\fgcolor{ff00ff}{=-cos\left(" + (If[a < 0, "-", ""]) + (Element[l, -abs(a) + 16]) + "\right)} ", ""]]) + "" + (LaTeX[If[h, "\fgcolor{ff00ff}{=-cos\left(" + (If[a ≤ 0, Element[l, a + 16], Element[l, a]]) + "\right)}", ""]]) + "" + (LaTeX[If[n, "\fgcolor{ff00ff}{=sin\left(" + (If[a < -1 * 4, Element[l, -4 - a], Element[l, 12 - a]]) + "\right)}", ""]]) + "" + (LaTeX[If[o, "\fgcolor{ff00ff}{=sin\left(" + (If[a ≤ 4, Element[l, a + 12], Element[l, abs(a) - 4]]) + "\right)}", ""]]) + "}"
texsin: "\fgcolor{ff0000}{sin\left(  " + (LaTeX[Element[l, a + 8]]) + "\right) = " + (LaTeX[Element[s, 8 + a]]) + "}" + (LaTeX[If[f_2 ∧ (a < 8), "\fgcolor{ff00ff}{=-sin\left(" + (Element[l, -a + 8]) + "\right)} ", ""]]) + "" + (LaTeX[If[g, "\fgcolor{ff00ff}{=sin\left(" + (If[a < 0, "-", ""]) + (Element[l, -abs(a) + 16]) + "\right)} ", ""]]) + "" + (LaTeX[If[h, "\fgcolor{ff00ff}{=-sin\left(" + (If[a ≤ 0, Element[l, a + 16], Element[l, a]]) + "\right)}", ""]]) + "" + (LaTeX[If[n, "\fgcolor{ff00ff}{=cos\left(" + (If[a < -1 * 4, Element[l, -4 - a], Element[l, 12 - a]]) + "\right)}", ""]]) + "" + (LaTeX[If[o, "\fgcolor{ff00ff}{=-cos\left(" + (If[a ≤ 4, Element[l, a + 12], Element[l, abs(a) - 4]]) + "\right)}", ""]]) + "}"