Sine Angle Addition Formula Proof
Author: John Maharry
Date: 05/28/09

Drag the points around the circle to change the angles. Sin(S+T) is the y-value of the point D. The value is also given by the sum of the two red (vertical) sides on the triangles shown to the right. These can be shown to be Sin(S)Cos(T) and Sin(T)Cos(S).

Sin(S)Cos(T)

Cos(T)

Sin(T)

g(x) = sin(x)
B = (0, 0)
C = (0, 1)
e = Circle(B, C)
A = Point(e)
D = Point(e)
E = Intersect(e, xAxis, 2)
a = Segment(B, A)
b = Segment(B, D)
c = OrthogonalLine(D, a)
F = Intersect(a, c)
d = OrthogonalLine(F, xAxis)
f = OrthogonalLine(D, d)
G = Intersect(d, f)
H = Intersect(d, xAxis)
h = Segment(D, F)
i = Segment(D, G)
j = Segment(G, F)
k = Segment(F, H)
I: (x(B) + 1.5, x(B) + 0.3)
J: (x(H) + 1.5, y(H) + 0.3)
K: (x(F) + 1.5, y(F) + 0.3)
L: (x(B) + 1.4, y(B) + 0.4)
M: (x(F) + 1.4, y(F) + 0.4)
N: (x(D) + 1.4, y(D) + 0.4)
O: (x(F) + 1.5, y(F) + 0.5)
P: (x(D) + 1.5, y(D) + 0.5)
Q: (x(G) + 1.5, y(G) + 0.5)
l = Segment(I, J)
m = Segment(J, K)
n = Segment(K, I)
p = Segment(L, M)
q = Segment(M, N)
ONE = Segment(N, L)
s = Segment(P, Q)
t = Segment(Q, O)
a_1 = Segment(O, P)
S = Angle(H, B, F)
T = Angle(F, B, D)
ST = Angle(H, B, D)
δ = Angle(J, I, K)
ε = Angle(M, L, N)
ζ = Angle(Q, O, P)
R = Midpoint(J, K)
U = Midpoint(Q, O)
V = Midpoint(N, L)
W = Midpoint(P, Q)
Y = Midpoint(I, J)
Z = Midpoint(K, I)
A_1 = Midpoint(O, P)
B_1: (x(R) + 0.1, y(R))
C_1: (x(Z) - 0.05, y(Z) + 0.05)