Author: 藍邦偉　老師

單位向量

座標軸

x

y

z

$ O $

$\sqrt3$

$1$

$2$

$1$

$\sqrt3$

$2$

$1$

$1$

$\sqrt2$

$s=%v$

$Case=%v$

廣義三角函數\;(3)

$30°$

\sqrt{2}

2

第二象限

第三象限

第四象限

$sin150°=\frac{\fgcolor{669900}1}{\fgcolor{blue}2}$
$cos150°=-\frac{\fgcolor{red}\sqrt3}{\fgcolor{blue}2}$
$tan150°=-\frac{\fgcolor{669900}1}{\fgcolor{red}\sqrt{3}}$

$sin210°=-\frac{\fgcolor{669900}1}{\fgcolor{blue}2}$
$cos210°=-\frac{\fgcolor{red}\sqrt3}{\fgcolor{blue}2}$
$tan210°=\frac{\fgcolor{669900}1}{\fgcolor{red}\sqrt{3}}$

$sin330°=-\frac{\fgcolor{669900}1}{\fgcolor{blue}2}$
$cos330°=\frac{\fgcolor{red}\sqrt3}{\fgcolor{blue}2}$
$tan330°=-\frac{\fgcolor{669900}1}{\fgcolor{red}\sqrt{3}}$

$sin135°=\frac{\fgcolor{669900}1}{\fgcolor{blue}\sqrt2}$
$cos135°=-\frac{\fgcolor{red}1}{\fgcolor{blue}\sqrt2}$
$tan135°=-\frac{\fgcolor{669900}1}{\fgcolor{red}1}=-1$

$sin225°=-\frac{\fgcolor{669900}1}{\fgcolor{blue}\sqrt2}$
$cos225°=-\frac{\fgcolor{red}1}{\fgcolor{blue}\sqrt2}$
$tan225°=\frac{\fgcolor{669900}1}{\fgcolor{red}1}=1$

$sin315°=-\frac{\fgcolor{669900}1}{\fgcolor{blue}\sqrt2}$
$cos315°=\frac{\fgcolor{red}1}{\fgcolor{blue}\sqrt2}$
$tan315°=-\frac{\fgcolor{669900}1}{\fgcolor{red}1}=-1$

$sin120°=\frac{\fgcolor{669900}\sqrt3}{\fgcolor{blue}2}$
$cos120°=-\frac{\fgcolor{red}1}{\fgcolor{blue}2}$
$tan120°=-\frac{\fgcolor{669900}\sqrt3}{\fgcolor{red}1}$

$sin240°=-\frac{\fgcolor{669900}\sqrt3}{\fgcolor{blue}2}$
$cos240°=-\frac{\fgcolor{red}1}{\fgcolor{blue}2}$
$tan240°=\frac{\fgcolor{669900}\sqrt3}{\fgcolor{red}1}$

$sin300°=-\frac{\fgcolor{669900}\sqrt3}{\fgcolor{blue}2}$
$cos300°=\frac{\fgcolor{red}1}{\fgcolor{blue}2}$
$tan300°=-\frac{\fgcolor{669900}\sqrt3}{\fgcolor{red}1}$

$150°$

$210°$

$330°$

$135°$

$225°$

$315°$

$120°$

$240°$

$300°$

2

2

\sqrt2

2

a

參考背景圖

t = 0.9 (slider from 0 to 1, step 0.01)
O = (10.925010619174007, 4.283447622340839)
α = 4.71238898038469 rad (slider from 0 to 6.283185307179586, step 1)
β = 1.5707963267948966 rad (slider from 0 to 6.283185307179586, step 1)
Y_{unit}: O + (-1 * sin(α), cos(α) * cos(β))
v = Vector(O, Y_{unit})
A_2: O + 2 * sqrt(3) * v
Z_{unit}: O + (0, sin(β))
w = Vector(O, Z_{unit})
A_1: O + 2 * sqrt(3) * v + 2 * w
X_{unit}: O + (cos(α), sin(α) * cos(β))
u = Vector(O, X_{unit})
L = 3 (slider from 1 to 10, step 1)
yy = Vector(O - L * v, O + L * v)
zz = Vector(O - L * w, O + L * w)
xx = Vector(O - L * u, O + L * u)
c = false
d = true
Q_2: 0.3 * Corner[2] + 0.7 * Corner[1]
Q_3: (x(Q_2), y(Corner[3]))
g = Segment(Q_2, Q_3)
polya, h, q_1, q_4, i = Polygon(Corner[1], Q_2, Q_3, Corner[4])
C_1: O + 2 * v + 2 * sqrt(3) * w
C_2: O + 2 * v
poly3, a, b, e = Polygon(C_1, C_2, O)
B_1: O + 2 * v + 2 * w
B_2: O + 2 * v
poly1, o, b_1, b_2 = Polygon(A_1, A_2, O)
poly2, o_1, c_1, c_2 = Polygon(B_1, B_2, O)
s = 0 (slider from 0 to 3, step 1)
B_3: O + 2 * v + 2 * sqrt(3) * w
x_{fake} = Vector((x(Q_2), y(O)), (x(Corner[2]), y(O)))
y_{fake} = Vector((x(O), y(Corner[2])), (x(O), y(Corner[3])))
case = 0 (slider from 0 to 3, step 1)
αmin = 1.5707963267948966 rad
αmax = 4.71238898038469 rad
βmin = 1.5707963267948966 rad
βmax = 4.71238898038469 rad
pl1 = PolyLine(O + 1.8 * sqrt(3) * v, O + 1.8 * sqrt(3) * v + 0.2 * sqrt(3) * w, O + 2 * sqrt(3) * v + 0.2 * sqrt(3) * w)
Angle1 = Angle(O + (1, 0), O, O + (sqrt(3), 1))
tri1, f, j, k = Polygon(O, O + (2 * sqrt(3), 2), O + (2 * sqrt(3), 0))
tric, l, m, n = Polygon(O, O + (2, 2 * sqrt(3)), O + (2, 0))
trib, p, o_2, q = Polygon(O, O + (2, 2), O + (2, 0))
A = (2.215615697839081, 7.202618664114236)
θ_{21} = Angle(O + (1, 0), O, O + (-sqrt(3), 1))
θ_{22} = Angle(O + (1, 0), O, O + (-sqrt(3), -1))
θ_{23} = Angle(O + (1, 0), O, O + (sqrt(3), -1))
θ_{31} = Angle(O + (1, 0), O, O + (-1, 1))
θ_{32} = Angle(O + (1, 0), O, O + (-1, -1))
θ_{33} = Angle(O + (1, 0), O, O + (1, -1))
θ_{41} = Angle(O + (1, 0), O, O + (-1, sqrt(3)))
θ_{42} = Angle(O + (1, 0), O, O + (-1, -sqrt(3)))
θ_{43} = Angle(O + (1, 0), O, O + (1, -sqrt(3)))
text13: "" + (LaTeX[If[t ≟ 1 ∧ (case ≟ 1 ∨ case ≟ 2) ∧ s ≟ 1, "-"]]) + "\sqrt3"
text12: "" + (LaTeX[If[t ≟ 1 ∧ (case ≟ 3 ∨ case ≟ 2) ∧ s ≟ 1, "-"]]) + "1"
text21: "" + (LaTeX[If[t ≟ 1 ∧ (case ≟ 1 ∨ case ≟ 2) ∧ s ≟ 2, "-"]]) + "1"
text22: "" + (LaTeX[If[t ≟ 1 ∧ (case ≟ 3 ∨ case ≟ 2) ∧ s ≟ 2, "-"]]) + "1"
text32: "" + (LaTeX[If[t ≟ 1 ∧ (case ≟ 3 ∨ case ≟ 2) ∧ s ≟ 3, "-"]]) + "\sqrt{3}"
text31: "" + (LaTeX[If[t ≟ 1 ∧ (case ≟ 1 ∨ case ≟ 2) ∧ s ≟ 3, "-"]]) + "1"
P_A: O + (4 * cos(t * π + π / 3), 4 * sin(t * π + π / 3))
P_B: O + (2 * cos(t * π), 2 * sin(t * π))
tria, p_b, o_3, p_a = Polygon(O, P_A, P_B)
Q_A: O + (2 * sqrt(2) * cos(t * π + π / 4), 2 * sqrt(2) * sin(t * π + π / 4))
Q_B: O + (2 * cos(t * π), 2 * sin(t * π))
tri2, q_b, o_4, q_a = Polygon(O, Q_A, Q_B)
R_A: O + (4 * cos(t * π + π / 6), 4 * sin(t * π + π / 6))
R_B: O + (2 * sqrt(3) * cos(t * π), 2 * sqrt(3) * sin(t * π))
tri3, r_b, o_5, r_a = Polygon(O, R_A, R_B)
text7: "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 1, "-"]]) + "1"
text9: "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 1, "-"]]) + "\sqrt3"
text5: "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 2, "-"]]) + "1"
text4: "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 2, "-"]]) + "1"
"1": "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 2, "-"]]) + "1"
text2: "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 3, "-"]]) + "1"
text3: "" + (LaTeX[If[t ≟ 1 ∧ case ≟ 2 ∧ s ≟ 3, "-"]]) + "\sqrt3"
φ_1 = Angle(P_A, P_B, O)
φ_2 = Angle(Q_A, Q_B, O)
φ_3 = Angle(R_A, R_B, O)
φ_a = Angle(O + (2 * sqrt(3), 1), O + (2 * sqrt(3), 0), O)
φ_b = Angle(O + (2, 1), O + (2, 0), O)
φ_c = Angle(O + (2, 1), O + (2, 0), O)
pl3 = PolyLine(O + 1.66 * v, O + 1.66 * v + 0.2 * sqrt(3) * w, O + 2 * v + 0.2 * sqrt(3) * w)
pl2 = PolyLine(O + 1.66 * v, O + 1.66 * v + 0.34 * w, O + 2 * v + 0.34 * w)
r = false
R_w: O + (2.5 * cos(t * π), 2.5 * sin(t * π))