隸美弗定理示例
Author: 藍邦偉  老師

$ 隸美弗定理示例 $

$ cos0°=1,sin0°=0 $

$ z=r\cdot (cosθ+isinθ) $
$⇒z^n=r^n\cdot [cos(nθ)+isin(nθ)]$

$cos15°=\frac{\sqrt{6}+\sqrt{2}}{4},sin15°=\frac{\sqrt{6}-\sqrt{2}}{4}  $

$ cos30°=\frac{1}{2} ,sin30°=\frac{\sqrt{3}}{2}$

$ cos45°=\frac{\sqrt{2}}{2},sin45=\frac{\sqrt{2}}{2} $

$ cos60°=\frac{1}{2},sin60°=\frac{\sqrt{3}}{2} $

$ cos75°=\frac{\sqrt{6}-\sqrt{2}}{4},sin75°=\frac{\sqrt{6}+\sqrt{2}}{4} $

$ cos90°=0,sin90°=1 $

$ cos105°=-\frac{\sqrt{6}-\sqrt{2}}{4},sin105°=\frac{\sqrt{6}+\sqrt{2}}{4} $

$ cos120°=-\frac{1}{2},sin120°=\frac{\sqrt{3}}{2} $

$ cos135°=-\frac{2}{2},sin135°=\frac{\sqrt{2}}{2} $

$ cos150°=-\frac{\sqrt{3}}{2},sin150°=\frac{1}{2} $

$ cos165°=-\frac{\sqrt{6}+\sqrt{2}}{4} ,sin165°=\frac{\sqrt{6}-\sqrt{2}}{4}$

$ cos180°=-1,sin180°=0 $

$ cos195°=-\frac{\sqrt{6}+\sqrt{2}}{4},sin195°=-\frac{\sqrt{6}-\sqrt{2}}{4} $

$ cos210°=-\frac{\sqrt{3}}{2},sin210°=-\frac{1}{2} $

$ cos225°=-\frac{\sqrt{2}}{2},sin225=-\frac{\sqrt{2}}{2} $

$ cos240°=-\frac{1}{2},sin240°=-\frac{\sqrt{3}}{2} $

$ cos255°=-\frac{\sqrt{6}-\sqrt{2}}{4} ,sin255°=-\frac{\sqrt{6}+\sqrt{2}}{4}$

$ cos270°=0,sin270°=-1 $

$ cos285°=\frac{\sqrt{6}-\sqrt{2}}{4},sin285°=-\frac{\sqrt{6}+\sqrt{2}}{4} $

$ cos300°=\frac{1}{2},sin300°=-\frac{\sqrt{3}}{2} $

$ cos315°=\frac{\sqrt{2}}{2},sin315°=-\frac{\sqrt{2}}{2} $

$ cos330°=\frac{\sqrt{3}}{2},sin330=-\frac{1}{2} $

$ cos345°=\frac{\sqrt{6}+\sqrt{2}}{4},sin345°=-\frac{\sqrt{6}-\sqrt{2}}{4} $

$ f(x)=sinx $

$ f(x)=cosx $

r = 2 (slider from 1 to 5, step 1)
n = 19 (slider from -30 to 30, step 1)
k = 11 (slider from 0 to 23, step 1)
text1: "$ z=" + (LaTeX[r]) + "\cdot (cos" + (LaTeX[15 * k]) + "°+isin" + (LaTeX[15 * k]) + "°) $"
text2: "$ z^{" + (LaTeX[n]) + "} =" + (LaTeX[r]) + "^{" + (LaTeX[n]) + "}\cdot (cos" + (LaTeX[15 * k]) + "°+isin" + (LaTeX[15 * k]) + "°)^{" + (LaTeX[n]) + "}$"
text3: "$ =" + (LaTeX[r^(n)]) + " \cdot (cos" + (LaTeX[15 * n * k]) + "°+isin" + (LaTeX[15 * n * k]) + "°)$"
text31: "$ =\frac{1}{" + (LaTeX[r]) + "^{" + (LaTeX[abs(n)]) + "}}\cdot [cos(" + (LaTeX[n * k * 15]) + "°)+isin(" + (LaTeX[n * k * 15]) + "°)]$"
α = Mod(15 * n * k, 360)
text4: "$ =" + (LaTeX[r^(n)]) + " \cdot (cos" + (LaTeX[α]) + "°+isin" + (LaTeX[α]) + "°)$"
text41: "$ =\frac{1}{" + (LaTeX[r^(-n)]) + "} \cdot [cos" + (LaTeX[α]) + "°+isin" + (LaTeX[α]) + "°]$"
Remark: "$ r=|z|=" + (LaTeX[r]) + ",θ=" + (LaTeX[k * 15]) + " °,n=" + (LaTeX[n]) + "$"
A = (7.82, 2.34)
f = Function(sin(x), 0, 6.283185307179586)
u = Vector((-1, 0), (7, 0))
v = Vector((8, 0), (16, 0))
w = Vector((0, -1.2), (0, 1.2))
z = Vector((9.42477796076938, -1.2), (9.42477796076938, -1.2))
a = Vector((9.42477796076938, -1.2), (9.42477796076938, 1.2))
b = Segment((1.5707963267948966, 0), (1.5707963267948966, 1))
c = Segment((4.71238898038469, 0), (4.71238898038469, -1))
g = Function(cos(x + π), 9.42478, 15.70796)
d = Segment((12.56637, 0), (12.56637, -1))
k_1 = Segment((15.707963267948966, 0), (15.707963267948966, 1))