加法公式の図解

作図からsinの加法公式を導出しよう。

\sin(\alpha+\beta)

O

P

B

H

Q

OP=1

OQ=\cos\alpha

K

QK=OQ\sin\beta=\cos\alpha\sin\beta

PQ=\sin\alpha

HQ=PQ\cos\beta=\sin\alpha\cos\beta

\sin(\alpha+\beta)=HK

=HQ+KQ

=\cos\alpha\sin\beta+\sin\alpha\cos\beta

O = (0, 0)
c = Circle(O, 1)
D = Intersect(c, xAxis, 1)
E = Intersect(c, xAxis, 2)
d = Line(D, E)
F = Intersect(c, yAxis, 2)
p = CircleArc(O, E, F)
A = Point(p)
A': ((-y(A)), x(A))
e = CircleArc(O, A, A')
B = Point(e)
α = Angle(A, O, B)
β = Angle(E, O, A)
a = Segment(O, A)
b = Segment(O, B)
f = OrthogonalLine(B, a)
C = Intersect(f, a)
g = Segment(B, C)
h = OrthogonalLine(C, d)
i = OrthogonalLine(B, h)
G = Intersect(h, i)
H = Intersect(d, h)
j = Segment(G, H)
I = Intersect(i, yAxis)
k = Segment(G, I)
γ = Angle(G, C, B)
J = Point(yAxis)
K = Point(yAxis)
u = Vector(J, K)
L = Point(d)
M = Point(d)
v = Vector(L, M)
l = LineBisector(I, O)
N = Point(l)
q = CircleArc(N, I, O)
m = 7 (slider from 0 to 7, step 1)