特別角的三角函數值(4)
Author: 藍邦偉  老師

$O$

$Step=%v$

$1$

1

1

x

36°

72°

72°

x

1-x

36°

 \frac{ x }{2 } 

考慮單位圓的內接正十邊形

$ 作一個單位圓 $

$ 特別角\;18°與\;72° $

$ 考慮單位圓的內接正十邊形 $

$ A $

$ B $

$ C $

$ D $

$ E $

$ F $

$ G $

H

$ I $

$ J $

$ 36° $

$ 72° $

$ 72° $

$ x $

$ 1 $

$ K $

$ 36° $

$ 36° $

1-x

$ x $

$ x $

$ ∵\Delta OAB≈\Delta ABK $

$ ∴\frac{1}{x}=\frac{x}{1-x} $

$ ⇒x²=1-x $

$ ⇒x²+x-1=0 $

$ ⇒x=\frac{-1±\sqrt{5}}{2} >0$

$ ⇒x=\frac{\sqrt{5}-1}{2} $

$step=%v$

$ M $

$ 18° $

$ \frac{x}{2} $

$ sin18°=cos72°=\frac{\frac{x}{2}}{1}=\frac{\sqrt{5}-1}{4} $
$cos18°=sin72°=\frac{\sqrt{1-(\frac{x}{2})^2}}{1}=\sqrt{1-(\frac{\sqrt{5}-1}{4})^2}=\frac{\sqrt{10+2\sqrt{5}}}{4}$

r: 0.4 * (y(Corner[4]) - y(Corner[1]))
O: 0.5 * Corner[1] + 0.5 * Corner[3] - (2, 0)
c = Circle(O, r)
n = 0 (slider from 0 to 6, step 1)
A: O + (r * cos(252°), r * sin(252°))
l = Segment(O, A)
D: O + (r * cos(0°), r * sin(0°))
C: O + (r * cos((-36)°), r * sin((-36)°))
E: O + (r * cos(36°), r * sin(36°))
F: O + (r * cos(72°), r * sin(72°))
G: O + (r * cos(108°), r * sin(108°))
H: O + (r * cos(144°), r * sin(144°))
I: O + (r * cos(180°), r * sin(180°))
J: O + (r * cos(216°), r * sin(216°))
B: O + (r * cos(288°), r * sin(288°))
a = Segment(A, B)
b = Segment(B, C)
d = Segment(C, D)
e = Segment(D, E)
f = Segment(E, F)
g = Segment(F, G)
h = Segment(G, H)
i = Segment(H, I)
j = Segment(I, J)
k = Segment(J, A)
m = Segment(O, B)
p, q_1 = AngularBisector(a, l)
K = Intersect(m, q_1)
s = Segment(A, K)
M = Midpoint(A, B)
t = Segment(O, M)
α = Angle(B, M, O)
q = 0 (slider from 0 to 6, step 1)
u = Vector(O + (0.5, -0.4), O + (0.05, -0.5))
L = (7.5, -12.16)