正弦倍角的證明
Author: 藍邦偉　老師

sin2θ的證明

ΔOPR中， \frac{  \overline{ PR }  }{sin(180°-2θ) } = \frac{  \overline{ OP }  }{sinθ } 

 \frac{  \overline{ PR }  }{sin2θ } = \frac{ 1  }{sinθ } 

sin2θ= \overline{ PR }sinθ 

2θ

θ

ΔOMR中，cosθ= \frac{  \overline{ MR }  }{ \overline{ OR }}= \frac{  \frac{  \overline{ PR }  }{2 }  }{  \overline{ OR } }= \frac{  \overline{ PR }  }{2 }     

 \overline{ PR }=2cosθ 

sin2θ=2sinθcosθ

n = 6 (slider from 0 to 6, step 1)
u = Vector((x(Corner[1.0]), 0.0), (x(Corner[3.0]), 0.0))
v = Vector((0.0, y(Corner[1.0])), (0.0, y(Corner[3.0])))
O = (0, 0)
α = 0.8726646259971648 rad (slider from 0 to 6.283185307179586, step 0.017453292519943295)
P: (cos(α), sin(α))
a = Segment(O, P)
R = (-1, 0)
b = Segment(P, R)
Q: (x(P), 0.0)
d = Segment(P, Q)
M = Midpoint(P, R)
e = Segment(O, M)
β = Angle(Q, O, P)
γ = Angle(O, R, P)
T_1 = Corner(1.0)
T_4 = Corner(4.0)
T_2: (6.0 * Corner[1.0] + 4.0 * Corner[2.0]) / 10.0
T_3: (6.0 * Corner[4.0] + 4.0 * Corner[3.0]) / 10.0
多邊形1, t_1, t_2, t_3, t_4 = Polygon(T_1, T_2, T_3, T_4)