特別角的三角函數值
Author: 藍邦偉  老師

$ 特別角的三角函數值 $

$ P $

$ A $

$ O $

$ sin\frac{π}{12}=sin15°=\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos\frac{π}{12}=cos15°=\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan\frac{π}{12}=tan15°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$
$cot\frac{π}{12}=cot15°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$
$sec\frac{π}{12}=sec15°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}$
$csc\frac{π}{12}=csc15°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}$

$sin\frac{π}{6}= sin30°=\frac{1}{2} $
$cos\frac{π}{6}=cos30°=\frac{\sqrt{3}}{2}$
$tan\frac{π}{6}=tan30°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$cot\frac{π}{6}=cot30°=\frac{\sqrt{3}}{1}=\sqrt{3}$
$sec\frac{π}{6}=sec30°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$
$csc\frac{π}{6}=csc30°=\frac{2}{1}=2$

$sin\frac{π}{4}=sin45°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} $
$cos\frac{π}{4}=cos45°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
$tan\frac{π}{4}=tan45°=\frac{1}{1}=1$
$cot\frac{π}{4}=cot45°=\frac{1}{1}=1$
$sec\frac{π}{4}=sec45°=\frac{\sqrt{2}}{1}=\sqrt{2}$
$csc\frac{π}{4}=csc45°=\frac{\sqrt{2}}{1}=\sqrt{2}$

$sin\frac{π}{3}=sin60°=\frac{\sqrt{3}}{2} $
$cos\frac{π}{3}=cos60°=\frac{1}{2}$
$tan\frac{π}{3}=tan60°=\frac{\sqrt{3}}{1}=\sqrt{3}$
$cot\frac{π}{3}=cot60°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$sec\frac{π}{3}=sec60°=\frac{2}{1}=2$
$csc\frac{π}{3}=csc60°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$

$sin\frac{5π}{12}=sin75°=\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos\frac{5π}{12}=cos75°=\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan\frac{5π}{12}=tan75°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$
$cot\frac{5π}{12}=cot75°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$
$sec\frac{5π}{12}=sec75°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}$
$csc\frac{5π}{12}=csc75°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}}$

$sin\frac{7π}{12}=sin105°=\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos\frac{7π}{12}=cos105°=-\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan\frac{7π}{12}=tan105°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}$
$cot\frac{7π}{12}=cot105°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}$
$sec\frac{7π}{12}=sec105°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}$
$csc\frac{7π}{12}=csc105°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}}$

$sin\frac{4π}{3}=sin240°=-\frac{\sqrt{3}}{2} $
$cos\frac{4π}{3}=cos240°=-\frac{1}{2}$
$tan\frac{4π}{3}=tan240°=\frac{\sqrt{3}}{1}=\sqrt{3}$
$cot\frac{4π}{3}=cot240°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$sec\frac{4π}{3}=sec240°=-\frac{2}{1}=-2$
$csc\frac{4π}{3}=csc240°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}$

$sin\frac{5π}{3}=sin300°=-\frac{\sqrt{3}}{2} $
$cos\frac{5π}{3}=cos300°=\frac{1}{2}$
$tan\frac{5π}{3}=tan300°=-\frac{\sqrt{3}}{1}=-\sqrt{3}$
$cot\frac{5π}{3}=cot300°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$
$sec\frac{5π}{3}=sec300°=\frac{2}{1}=2$
$csc\frac{5π}{3}=csc300°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}$

$sin\frac{3π}{4}=sin135°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2} $
$cos\frac{3π}{4}=cos135°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}$
$tan\frac{3π}{4}=tan135°=-\frac{1}{1}=-1$
$cot\frac{3π}{4}=cot135°=-\frac{1}{1}=-1$
$sec\frac{3π}{4}=sec135°=-\frac{\sqrt{2}}{1}=-\sqrt{2}$
$csc\frac{3π}{4}=csc135°=\frac{\sqrt{2}}{1}=\sqrt{2}$

$sin\frac{5π}{4}=sin225°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2} $
$cos\frac{5π}{4}=cos225°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2}$
$tan\frac{5π}{4}=tan225°=\frac{1}{1}=1$
$cot\frac{5π}{4}=cot225°=\frac{1}{1}=1$
$sec\frac{5π}{4}=sec225°=-\frac{\sqrt{2}}{1}=-\sqrt{2}$
$csc\frac{5π}{4}=csc225°=-\frac{\sqrt{2}}{1}=-\sqrt{2}$

$sin\frac{7π}{4}=sin315°=-\frac{1}{\sqrt{2}}=-\frac{\sqrt{2}}{2} $
$cos\frac{7π}{4}=cos315°=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}$
$tan\frac{7π}{4}=tan315°=-\frac{1}{1}=-1$
$cot\frac{7π}{4}=cot315°=-\frac{1}{1}=-1$
$sec\frac{7π}{4}=sec315°=\frac{\sqrt{2}}{1}=\sqrt{2}$
$csc\frac{7π}{4}=csc315°=-\frac{\sqrt{2}}{1}=-\sqrt{2}$

$sin\frac{5π}{6}=sin150°=\frac{1}{2} $
$cos\frac{5π}{6}=cos150°=-\frac{\sqrt{3}}{2}$
$tan\frac{5π}{6}=tan150°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$
$cot\frac{5π}{6}=cot150°=-\frac{\sqrt{3}}{1}=-\sqrt{3}$
$sec\frac{5π}{6}=sec150°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}$
$csc\frac{5π}{6}=csc150°=\frac{2}{1}=2$

$sin\frac{7π}{6}=sin210°=-\frac{1}{2} $
$cos\frac{7π}{6}=cos210°=-\frac{\sqrt{3}}{2}$
$tan\frac{7π}{6}=tan210°=\frac{1}{\sqrt{3}}=\frac{\sqrt{3}}{3}$
$cot\frac{7π}{6}=cot210°=\frac{\sqrt{3}}{1}=\sqrt{3}$
$sec\frac{7π}{6}=sec210°=-\frac{2}{\sqrt{3}}=-\frac{2\sqrt{3}}{3}$
$csc\frac{7π}{6}=csc210°=-\frac{2}{1}=-2$

$sin\frac{11π}{6}=sin330°=-\frac{1}{2} $
$cos\frac{11π}{6}=cos330°=\frac{\sqrt{3}}{2}$
$tan\frac{11π}{6}=tan330°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$
$cot\frac{11π}{6}=cot330°=-\frac{\sqrt{3}}{1}=-\sqrt{3}$
$sec\frac{11π}{6}=sec330°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$
$csc\frac{11π}{6}=csc330°=-\frac{2}{1}=-2$

$sin\frac{11π}{12}=sin165°=\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos\frac{11π}{12}=cos165°=-\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan\frac{11π}{12}=tan165°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}$
$cot\frac{11π}{12}=cot165°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}$
$sec\frac{11π}{12}=sec165°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}$
$csc\frac{11π}{12}=csc165°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}$

$sin\frac{13π}{12}=sin195°=-\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos\frac{13π}{12}=cos195°=-\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan\frac{13π}{12}=tan195°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$
$cot\frac{13π}{12}=cot195°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$
$sec\frac{13π}{12}=sec195°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}$
$csc\frac{13π}{12}=csc195°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}$

$sin\frac{23π}{12}=sin345°=-\frac{\sqrt{6}-\sqrt{2}}{4} $
$cos\frac{23π}{12}=cos345°=\frac{\sqrt{6}+\sqrt{2}}{4}$
$tan\frac{23π}{12}=tan345°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}$
$cot\frac{23π}{12}=cot345°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}$
$sec\frac{23π}{12}=sec345°=\frac{4}{\sqrt{6}+\sqrt{2}}=\sqrt{6}-\sqrt{2}$
$csc\frac{23π}{12}=csc345°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}$

$sin\frac{17π}{12}=sin255°=-\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos\frac{17π}{12}=cos255°=-\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan\frac{17π}{12}=tan255°=\frac{2+\sqrt{3}}{1}=2+\sqrt{3}$
$cot\frac{17π}{12}=cot255°=\frac{1}{2+\sqrt{3}}=2-\sqrt{3}$
$sec\frac{17π}{12}=sec255°=-\frac{4}{\sqrt{6}-\sqrt{2}}=-\sqrt{6}-\sqrt{2}$
$csc\frac{17π}{12}=csc255°=-\frac{4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}}$

$sin\frac{19π}{12}=sin285°=-\frac{\sqrt{6}+\sqrt{2}}{4} $
$cos\frac{19π}{12}=cos285°=\frac{\sqrt{6}-\sqrt{2}}{4}$
$tan\frac{19π}{12}=tan285°=-\frac{2+\sqrt{3}}{1}=-2-\sqrt{3}$
$cot\frac{19π}{12}=cot285°=-\frac{1}{2+\sqrt{3}}=-2+\sqrt{3}$
$sec\frac{19π}{12}=sec285°=\frac{4}{\sqrt{6}-\sqrt{2}}=\sqrt{6}+\sqrt{2}$
$csc=\frac{19π}{12}=csc285°=\frac{-4}{\sqrt{6}+\sqrt{2}}=-\sqrt{6}+\sqrt{2}}$

$ sin\frac{3π}{2}=sin270°=-1 $
$ cos\frac{3π}{2}=cos270°=0 $
$ tan\frac{3π}{2}=tan270°\;無意義 $
$ cot\frac{3π}{2}=cot270°=0 $
$ sec\frac{3π}{2}=sec270°\;無意義 $
$ csc\frac{3π}{2}=csc270°=-1 $

$ sinπ=sin180°=0 $
$ cosπ=cos180°=-1 $
$ tanπ=tan180°=0 $
$ cotπ=cot180°\;無意義 $
$ secπ=sec180°=-1 $
$ cscπ=csc180°\; 無意義$

$ sin\frac{π}{2}=sin90°=1 $
$cos\frac{π}{2}=cos90°=0$
$tan\frac{π}{2}=tan90°\;無意義$
$cot\frac{π}{2}=cot90°=0$
$sec\frac{π}{2}=sec90°\;無意義$
$csc\frac{π}{2}=csc90°=1$

$sin\frac{2π}{3}=sin120°=\frac{\sqrt{3}}{2} $
$cos\frac{2π}{3}=cos120°=-\frac{1}{2}$
$tan\frac{2π}{3}=tan120°=-\frac{\sqrt{3}}{1}=-\sqrt{3}$
$cot\frac{2π}{3}=cot120°=-\frac{1}{\sqrt{3}}=-\frac{\sqrt{3}}{3}$
$sec\frac{2π}{3}=sec120°=-\frac{2}{1}=-2$
$csc\frac{2π}{3}=csc120°=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}$

$sin0=sin0°=0$
$cos0=cos0°=1$
$tan0=tan0°=0$
$cot0=cot0°\;無意義$
$sec0=sec0°=-1$
$csc0=csc0°\;無意義$

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: 0.53 * Corner[1] + 0.47 * 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 = (-4.1819954724708905, 1.0309352842253556)
β = Angle(P, A, O)