特別角的三角函數值
Author: 藍邦偉  老師

$Case=%v$

$ 特別角的三角函數值 $

$ P $

$ A $

$ O $

$%v$

$ sin\;0°=0 $
$\;$
$cos\;0°=1$
$\;$
$tan\;0°=0$
$\;$
$cot\;0°=無意義$
$\;$
$sec\;0°=1$
$\;$
$csc\;0°=無意義$

$  sin\;270°=-1$
$\;$
$cos\;270°=0$
$\;$
$tan\;270°=無意義$
$\;$
$cot\;270°=0$
$\;$
$sec\;270°=無意義$
$\;$
$csc\;270°=-1$

$ sin180°=0 $
$\;$
$cos180°=-1$
$\;$
$tan180°=0$
$\;$
$cot180°=無意義$
$\;$
$sec180°=-1$
$\;$
$csc180°=無意義$

$ sin\;90°=1 $
$\;$
$cos\;90°=0$
$\;$
$tan\;90°=無意義$
$\;$
$cot\;90°=0$
$\;$
$sec\;90=無意義$
$\;$
$csc\;90°=1$

顯示\;cotθ,secθ,cscθ

u = Vector((x(Corner[1]), 0), (x(Corner[3]), 0))
v = Vector((0, y(Corner[1])), (0, y(Corner[3])))
O = (0, 0)
Q_2: 0.6 * Corner[1] + 0.4 * Corner[2]
Q_3: (x(Q_2), y(Corner[4]))
poly1, q_1, q_2, q_4, q_3 = Polygon(Corner[1], Q_2, Q_3, Corner[4])
c = Circle(O, 1)
list_1: {0*°, 15*°, 30*°, 45*°, 60*°, 75*°, 90*°, 105*°, 120*°, 135*°, 150*°, 165*°, 180*°, 195*°, 210*°, 225*°, 240*°, 255*°, 270*°, 285*°, 300*°, 315*°, 330*°, 345*°}
n = 0 (slider from 0 to 23, step 1)
P: (cos(Element[list_1, n + 1]), sin(Element[list_1, n + 1]))
A: (x(P), 0)
a = Segment(P, A)
b = Segment(P, O)
d = Segment(O, A)
S = (1, 0)
α = Angle(S, O, P)
B = (-3.532126427174519, 1.0716295617600593)
β = Angle(P, A, O)
e = false
text01: "$ sin15°=\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos15°=\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan15°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot15°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec15°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc15°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}"]]) + "$"
text02: "$ sin30°=\frac{1}{2} $
$cos30°=\frac{\sqrt{3}}{2}$
$tan30°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$" + (LaTeX[If[e ≟ true, "cot30°=\frac{\sqrt{3}}{1}=\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec30°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc30°=\frac{2}{1}=2"]]) + "$"
text23: "$ sin345°=-\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos345°=\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan345°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot345°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec345°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc345°=-\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}"]]) + "$"
text21: "$sin315°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2} $
$cos315°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
$tan315°=-\frac{1}{1}=-1$
$" + (LaTeX[If[e ≟ true, "cot315°=-\frac{1}{1}=-1"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec315°=\frac{\sqrt{2}}{1}=\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc315°=-\frac{\sqrt{2}}{1}=-\sqrt{2}"]]) + "$"
text20: "$sin300°=-\frac{\sqrt{3}}{2} $
$cos300°=\frac{1}{2}$
$tan300°=-\frac{\sqrt{3}}{1}=-\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot300°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec300°=\frac{2}{1}=2"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc300°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}"]]) + "$"
text17: "$ sin255°=-\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos255°=-\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan255°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot255°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec255°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc255°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}}"]]) + "$"
text16: "$ sin240°=-\frac{\sqrt{3}}{2} $
$cos240°=-\frac{1}{2}$
$tan240°=\frac{\sqrt{3}}{1}=\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot240°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec240°=-\frac{2}{1}=-2"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc240°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}"]]) + "$"
text15: "$sin225°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2} $
$cos225°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}$
$tan225°=\frac{1}{1}=1$
$" + (LaTeX[If[e ≟ true, "cot225°=\frac{1}{1}=1"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec225°=-\frac{\sqrt{2}}{1}=-\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc225°=-\frac{\sqrt{2}}{1}=-\sqrt{2}"]]) + "$"
text14: "$ sin210°=-\frac{1}{2} $
$cos210°=-\frac{\sqrt{3}}{2}$
$tan210°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$" + (LaTeX[If[e ≟ true, "cot210°=\frac{\sqrt{3}}{1}=\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec210°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc210°=-\frac{2}{1}=-2"]]) + "$"
text13: "$ sin195°=-\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos195°=-\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan195°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot195°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec195°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc195°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}"]]) + "$"
text11: "$ sin165°=\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos165°=-\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan165°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot165°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec165°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc165°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}"]]) + "$"
text10: "$ sin150°=\frac{1}{2} $
$cos150°=-\frac{\sqrt{3}}{2}$
$tan150°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$
$" + (LaTeX[If[e ≟ true, "cot150°=-\frac{\sqrt{3}}{1}=-\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec150°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc150°=\frac{2}{1}=2"]]) + "$"
text09: "$sin135°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} $
$cos135°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}$
$tan135°=-\frac{1}{1}=-1$
$" + (LaTeX[If[e ≟ true, "cot135°=-\frac{1}{1}=-1"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec135°=-\frac{\sqrt{2}}{1}=-\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc135°=\frac{\sqrt{2}}{1}=\sqrt{2}"]]) + "$"
text08: "$ sin120°=\frac{\sqrt{3}}{2} $
$cos120°=-\frac{1}{2}$
$tan120°=-\frac{\sqrt{3}}{1}=-\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot120°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec120°=-\frac{2}{1}=-2"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc120°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}"]]) + "$"
text07: "$ sin105°=\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos105°=-\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan105°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot105°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec105°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc105°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}}"]]) + "$"
text05: "$ sin75°=\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos75°=\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan75°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot75°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec75°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc75°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}}"]]) + "$"
text04: "$ sin60°=\frac{\sqrt{3}}{2} $
$cos60°=\frac{1}{2}$
$tan60°=\frac{\sqrt{3}}{1}=\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot60°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec60°=\frac{2}{1}=2"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc60°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}"]]) + "$"
text03: "$ sin45°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} $
$cos45°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
$tan45°=\frac{1}{1}=1$
$" + (LaTeX[If[e ≟ true, "cot45°=\frac{1}{1}=1"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec45°=\frac{\sqrt{2}}{1}=\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc45°=\frac{\sqrt{2}}{1}=\sqrt{2}"]]) + "$"
poly_q, f, g, h, i = Polygon(B + (0, -0.8), B + (1.7, -0.8), B + (1.7, -2), B - (0, 2))
text22: "$ sin330°=-\frac{1}{2} $
$cos330°=\frac{\sqrt{3}}{2}$
$tan330°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$
$" + (LaTeX[If[e ≟ true, "cot330°=-\frac{\sqrt{3}}{1}=-\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec330°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc330°=-\frac{2}{1}=-2"]]) + "$"
text19: "$ sin285°=-\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos285°=\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan285°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}$
$" + (LaTeX[If[e ≟ true, "cot285°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}"]]) + "$
$" + (LaTeX[If[e ≟ true, "sec285°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}"]]) + "$
$" + (LaTeX[If[e ≟ true, "csc285°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}}"]]) + "$"
tri, p, o, a_1 = Polygon(O, A, P)