Trigonometriskās funkcijas

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Trigonometrisko funkciju grafiki: sinuss, kosinuss, tangenss, kotangenss, sekanss, kosekanss

Trigonometriska funkcija ir jebkura no funkcijām sin x, cos x, tg x, ctg x, sec x un cosec x, kur arguments x ir leņķis. Raksturīga šo funkciju īpašība ir to periodiskums.
Ne katra periodiska funkcija, kuras arguments ir leņķis, ir trigonometriska funkcija. Piemēram, funkcija e^{\sin x} + \cos x nav trigonometriska funkcija.

Trigonometrisko funkciju uzskaitījums[izmainīt šo sadaļu | labot pirmkodu]

Vienības aplis ar kosinusa un sinusa vērtībām
Funkcija Apzīmējums Apraksts Sakarības (izmantojot radiānus)
Sinuss sin \frac {\textrm{pretkatete}} {\textrm{hipoten}\bar{\textrm{u}}\textrm{za}} \sin \theta \equiv \cos \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\csc \theta}
Kosinuss cos \frac {\textrm{piekatete}} {\textrm{hipoten}\bar{\textrm{u}}\textrm{za}} \cos \theta \equiv \sin \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\sec \theta}\,
Tangenss tg \frac {\textrm{pretkatete}} {\textrm{piekatete}} \operatorname{tg} \theta \equiv \frac{\sin \theta}{\cos \theta} \equiv \cot \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\operatorname{ctg} \theta}
Kotangenss ctg \frac {\textrm{piekatete}} {\textrm{pretkatete}} \operatorname{ctg} \theta \equiv \frac{\cos \theta}{\sin \theta} \equiv \operatorname{tg} \left(\frac{\pi}{2} - \theta \right) \equiv \frac{1}{\operatorname{tg} \theta}
Sekanss sec \frac {\textrm{hipoten}\bar{\textrm{u}}\textrm{za}} {\textrm{piekatete}} \sec \theta \equiv \csc \left(\frac{\pi}{2} - \theta \right) \equiv\frac{1}{\cos \theta}
Kosekanss cosec
(vai csc)
\frac {\textrm{hipoten}\bar{\textrm{u}}\textrm{za}} {\textrm{pretkatete}} \csc \theta \equiv \sec \left(\frac{\pi}{2} - \theta \right) \equiv\frac{1}{\sin \theta}

Trigonometrisko funkciju galveno vērtību tabula[izmainīt šo sadaļu | labot pirmkodu]

 \alpha \,\! 0° (0 rad) 30° (π/6) 45° (π/4) 60° (π/3) 90° (π/2) 180° (π) 270° (3π/2) 360° (2π)
 \sin \alpha \,\! {0} \,\!  \frac{1}{2}\,\!  \frac{ \sqrt{2}}{2}\,\!  \frac{ \sqrt{3}}{2}\,\! {1}\,\! {0}\,\! {-1}\,\! {0}\,\!
 \cos \alpha \,\! {1} \,\!   \frac{ \sqrt{3}}{2}\,\!  \frac{ \sqrt{2}}{2}\,\!  \frac{1}{2}\,\! {0}\,\! {-1}\,\! {0}\,\! {1}\,\!
 \mathop{\mathrm{tg}}\, \alpha \,\! {0} \,\!  \frac{1}{ \sqrt{3}}\,\!  {1}\,\!   \sqrt{3}\,\! \infin {0}\,\! \infin {0}\,\!
 \mathop{\mathrm{ctg}}\, \alpha \,\! \infin   \sqrt{3}\,\! {1} \,\!  \frac{1}{ \sqrt{3}}\,\!  {0}\,\! \infin {0}\,\! \infin
 \sec \alpha \,\! {1} \,\!   \frac{2}{ \sqrt{3}}\,\!   \sqrt{2}\,\!  {2}\,\! \infin {-1}\,\! \infin  {1}\,\!
 \operatorname{cosec}\, \alpha \,\! \infin  {2}\,\!   \sqrt{2}\,\!  \frac{2}{ \sqrt{3}}\,\! {1}\,\! \infin {-1}\,\! \infin

Trigonometrisko funkciju vērtības nestandarta leņķiem[izmainīt šo sadaļu | labot pirmkodu]

\alpha\, \frac{\pi}{12} = 15^\circ \frac{\pi}{10} = 18^\circ \frac{\pi}{8} = 22,5^\circ \frac{\pi}{5} = 36^\circ \frac{3\,\pi}{10} = 54^\circ \frac{3\,\pi}{8} = 67,5^\circ \frac{2\,\pi}{5} = 72^\circ
\sin \alpha\, \frac{\sqrt{3}-1}{2\,\sqrt{2}} \frac{\sqrt{5}-1}{4} \frac{\sqrt{2-\sqrt{2}}}{2} \frac{\sqrt{5-\sqrt{5}}}{2\,\sqrt{2}} \frac{\sqrt{5}+1}{4} \frac{\sqrt{2+\sqrt{2}}}{2} \frac{\sqrt{5+\sqrt{5}}}{2\,\sqrt{2}}
\cos \alpha\, \frac{\sqrt{3}+1}{2\,\sqrt{2}} \frac{\sqrt{5+\sqrt{5}}}{2\,\sqrt{2}} \frac{\sqrt{2+\sqrt{2}}}{2} \frac{\sqrt{5}+1}{4} \frac{\sqrt{5-\sqrt{5}}}{2\,\sqrt{2}} \frac{\sqrt{2-\sqrt{2}}}{2} \frac{\sqrt{5}-1}{4}
\operatorname{tg}\,\alpha 2-\sqrt{3} \sqrt{1-\frac{2}{\sqrt{5}}} \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}} \sqrt{5-2\,\sqrt{5}} \sqrt{1+\frac{2}{\sqrt{5}}} \sqrt{\frac{\sqrt{2}+1}{\sqrt{2}-1}} \sqrt{5+2\,\sqrt{5}}
\operatorname{ctg}\,\alpha 2 + \sqrt{3} \sqrt{5+2\,\sqrt{5}} \sqrt{\frac{\sqrt{2}+1}{\sqrt{2}-1}} \sqrt{1+\frac{2}{\sqrt{5}}} \sqrt{5-2\,\sqrt{5}} \sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}} \sqrt{1-\frac{2}{\sqrt{5}}}

\operatorname{tg} \frac{\pi}{120}= \operatorname{tg} 1,5^\circ =\sqrt{\frac{8-\sqrt{2(2-\sqrt{3})(3-\sqrt{5})} - \sqrt{
2(2+\sqrt{3})(5+\sqrt{5})}}{8+\sqrt{2(2-\sqrt{3})(3-\sqrt{5})}+\sqrt{2(2+\sqrt{3})(5+\sqrt{5})}
}}

\cos \frac{\pi}{240}=\frac{1}{16}\left(\sqrt{2-k} \left(\sqrt{2(5+\sqrt{5})}+\sqrt{3}-\sqrt{15} \right) + \sqrt{2+k} \left (\sqrt{6(5+\sqrt{5})}+\sqrt{5} - 1 \right) \right), kur k=\sqrt{2+\sqrt{2}}.

\cos \frac{\pi}{17} = \frac{1}{8}\sqrt{2 \left(2\sqrt{\sqrt{\frac{17k}{2}}
-\sqrt{\frac{k}{2}}-4\sqrt{2(17+\sqrt{17})} + 3\sqrt{17}+17}+\sqrt{2k}+\sqrt{17}+15 \right)}, kur k = 17 - \sqrt{17}.

Trigonometrisko funkciju īpašības[izmainīt šo sadaļu | labot pirmkodu]

Galvenās vienādības[izmainīt šo sadaļu | labot pirmkodu]

Tā kā sinuss un kosinuss ir attiecīgi punkta ordināta un abscisa, kas atbilst leņķa α riņķim, tad, atbilstoši Pitagora teorēmai

 \sin^2 \alpha + \cos^2 \alpha = 1. \qquad \qquad \,

Dalot šīs vienādības abas puses ar sinusa kvadrātu vai kosinusa kvadrātu, iegūstam:

 1 + \mathop{\mathrm{tg}}\,^2 \alpha = \frac{1}{ \cos^2 \alpha}, \qquad \qquad  \,
 1 + \mathop{\mathrm{ctg}}\,^2 \alpha = \frac{1}{ \sin^2 \alpha}. \qquad \qquad  \,

Nepārtrauktība[izmainīt šo sadaļu | labot pirmkodu]

Sinuss un kosinuss ir nepārtrauktas funkcijas, bet tangensam, kotangensam, sekansam un kosekansam ir pārtraukuma punkti \pm\frac{\pi}{2},\;\pm\pi,\;\pm\frac{3\pi}{2},\;\dots kotangenss un kosekanss — 0,\;\pm\pi,\;\pm2\pi,\;\dots

Paritāte[izmainīt šo sadaļu | labot pirmkodu]

Kosinuss un sekanss ir funkcijas, kurām ir simetrija attiecībā uz funkcijas zīmes maiņu. Pārējām četrām funkcijām tādas īpašības nav, t.i.:

 \sin \left( - \alpha \right)  = - \sin \alpha \,,
 \cos \left( - \alpha \right)  =  \cos \alpha \,,
 \mathop{\mathrm{tg}}\, \left( - \alpha \right)  = - \mathop{\mathrm{tg}}\, \alpha \,,
 \mathop{\mathrm{ctg}}\, \left( - \alpha \right)  = - \mathop{\mathrm{ctg}}\, \alpha \,,
 \sec \left( - \alpha \right)  =  \sec \alpha \,,
 \mathop{\mathrm{cosec}}\, \left( - \alpha \right)  = - \mathop{\mathrm{cosec}}\, \alpha \,.

Periodiskums[izmainīt šo sadaļu | labot pirmkodu]

Funkcijas y = sin α, y = cos α, y = sec α, y = cosec α — periodiskas funkcijas ar periodu . Funkcijas y = tg α, y = ctg α — ar periodu π

Saskaitīšanas formulas[izmainīt šo sadaļu | labot pirmkodu]

Summas trigonometriskās funkcijas nozīme un divu leņķu starpība:

 \sin\left( \alpha \pm \beta \right)= \sin\alpha \, \cos\beta \pm \cos\alpha \, \sin\beta,
 \cos\left( \alpha \pm \beta \right)= \cos\alpha \, \cos\beta \mp \sin\alpha \, \sin\beta,
 \operatorname{tg}\left( \alpha \pm \beta \right) = \frac{\operatorname{tg}\,\alpha \pm \operatorname{tg}\,\beta}{1 \mp \operatorname{tg}\,\alpha \, \operatorname{tg}\,\beta},
 \operatorname{ctg}\left( \alpha \pm \beta \right) = \frac{\operatorname{ctg}\,\alpha\,\operatorname{ctg}\,\beta \mp 1}{\operatorname{ctg}\,\beta \pm \operatorname{ctg}\,\alpha}.

Līdzīgas formulas trim leņķiem:

\sin \left( \alpha + \beta + \gamma \right) = \sin \alpha \cos \beta \cos \gamma + \cos \alpha \sin \beta \cos \gamma + \cos \alpha \cos \beta \sin \gamma - \sin \alpha \sin \beta \sin \gamma,
\cos \left( \alpha + \beta + \gamma \right) = \cos \alpha \cos \beta \cos \gamma - \sin \alpha \sin \beta \cos \gamma - \sin \alpha \cos \beta \sin \gamma - \cos \alpha \sin \beta \sin \gamma.

Formulas leņķu daudzkārtņiem[izmainīt šo sadaļu | labot pirmkodu]

Divkārša leņķa formulas:

\sin 2\alpha = 2 \sin \alpha \cos \alpha = \frac{2\,\operatorname{tg}\,\alpha }{1 + \operatorname{tg}^2\alpha},
\cos 2\alpha = \cos^2 \alpha\,-\,\sin^2 \alpha = 2 \cos^2 \alpha\,-\,1 = 1\,-\,2 \sin^2 \alpha = \frac{1 - \operatorname{tg}^2 \alpha}{1 + \operatorname{tg}^2\alpha} = \frac{\operatorname{ctg}\,\alpha - \operatorname{tg}\,\alpha}{\operatorname{ctg}\,\alpha + \operatorname{tg}\,\alpha},
\operatorname{tg}\,2 \alpha = \frac{2\,\operatorname{tg}\,\alpha}{1 - \operatorname{tg}^2\alpha},
\operatorname{ctg}\,2 \alpha = \frac{\operatorname{ctg}^2 \alpha - 1}{2\,\operatorname{ctg}\,\alpha} = \frac{1}{2}\left(\operatorname{ctg}\,\alpha - \operatorname{tg}\,\alpha \right).

Trīskārša leņķa formulas:

\sin\,3\alpha=3\sin\alpha - 4\sin^3\alpha,
\cos\,3\alpha=4\cos^3\alpha -3\cos\alpha,
\operatorname{tg}\,3\alpha=\frac{3\,\operatorname{tg}\,\alpha - \operatorname{tg}^3\,\alpha}{1 - 3\,\operatorname{tg}^2\,\alpha},
\operatorname{ctg}\,3\alpha=\frac{\operatorname{ctg}^3\,\alpha - 3\,\operatorname{ctg}\,\alpha}{3\,\operatorname{ctg}^2\,\alpha - 1}.

Citas leņķu daudzkārtņu formulas:

\sin\,4\alpha=\cos\alpha \left(4\sin\alpha - 8\sin^3\alpha\right),
\cos\,4\alpha=8\cos^4\alpha - 8\cos^2\alpha + 1,
\operatorname{tg}\,4\alpha=\frac{4\,\operatorname{tg}\,\alpha - 4\,\operatorname{tg}^3\,\alpha}{1 - 6\,\operatorname{tg}^2\,\alpha + \operatorname{tg}^2\,\alpha},
\operatorname{ctg}\,4\alpha=\frac{\operatorname{ctg}^4\,\alpha - 6\,\operatorname{ctg}^2\,\alpha + 1}{4\,\operatorname{ctg}^3\,\alpha - 4\,\operatorname{ctg}\,\alpha},
\sin\,5\alpha=16\sin^5\alpha-20\sin^3\alpha +5\sin\alpha
\cos\,5\alpha=16\cos^5\alpha-20\cos^3\alpha +5\cos\alpha
\operatorname{tg}\,5\alpha=\operatorname{tg}\alpha\frac{\operatorname{tg}^4\alpha-10\operatorname{tg}^2\alpha+5}{5\operatorname{tg}^4\alpha-10\operatorname{tg}^2\alpha+1}
 \sin (n\alpha)=2^{n-1}\prod^{n-1}_{k=0}\sin\left( \alpha+\frac{\pi k}{n}\right)

Pusleņķa formulas:

\sin\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{2}},\quad 0 \leqslant \alpha \leqslant 2\pi,
\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}},\quad -\pi \leqslant \alpha \leqslant \pi,
\operatorname{tg}\,\frac{\alpha}{2}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha},
\operatorname{ctg}\,\frac{\alpha}{2}=\frac{\sin\alpha}{1-\cos\alpha}=\frac{1+\cos\alpha}{\sin\alpha},
\operatorname{tg}\,\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}},\quad 0 \leqslant \alpha < \pi,
\operatorname{ctg}\,\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}},\quad 0 < \alpha \leqslant \pi.

Reizināšana[izmainīt šo sadaļu | labot pirmkodu]

Formulas divu leņķu reizināšanai:

\sin\alpha \sin\beta = \frac{\cos(\alpha-\beta) - \cos(\alpha+\beta)}{2},
\sin\alpha \cos\beta = \frac{\sin(\alpha-\beta) + \sin(\alpha+\beta)}{2},
\cos\alpha \cos\beta = \frac{\cos(\alpha-\beta) + \cos(\alpha+\beta)}{2},
\operatorname{tg}\,\alpha\,\operatorname{tg}\,\beta = \frac{\cos(\alpha-\beta) - \cos(\alpha+\beta)}{\cos(\alpha-\beta) + \cos(\alpha+\beta)},
\operatorname{tg}\,\alpha\,\operatorname{ctg}\,\beta = \frac{\sin(\alpha-\beta) + \sin(\alpha+\beta)}{\sin(\alpha+\beta) -\sin(\alpha-\beta)},
\operatorname{ctg}\,\alpha\,\operatorname{ctg}\,\beta = \frac{\cos(\alpha-\beta) + \cos(\alpha+\beta)}{\cos(\alpha-\beta) - \cos(\alpha+\beta)}.

Līdzīgas formulas triju leņķu sinusu un kosinusu reizināšanai:

\sin\alpha \sin\beta \sin\gamma = \frac{\sin(\alpha+\beta-\gamma) + \sin(\beta+\gamma-\alpha) + \sin(\alpha-\beta+\gamma) - \sin(\alpha+\beta+\gamma)}{4},
\sin\alpha \sin\beta \cos\gamma = \frac{-\cos(\alpha+\beta-\gamma) + \cos(\beta+\gamma-\alpha) + \cos(\alpha-\beta+\gamma) - \cos(\alpha+\beta+\gamma)}{4},
\sin\alpha \cos\beta \cos\gamma = \frac{\sin(\alpha+\beta-\gamma) - \sin(\beta+\gamma-\alpha) + \sin(\alpha-\beta+\gamma) - \sin(\alpha+\beta+\gamma)}{4},
\cos\alpha \cos\beta \cos\gamma = \frac{\cos(\alpha+\beta-\gamma) + \cos(\beta+\gamma-\alpha) + \cos(\alpha-\beta+\gamma) + \cos(\alpha+\beta+\gamma)}{4}.

Attiecīgās formulas triju leņķu tangensiem un kotangensiem var iegūt, izdalot augstāk minēto vienādojumu labās puses ar kreisajām.

Pakāpes[izmainīt šo sadaļu | labot pirmkodu]

\sin^2\alpha = \frac{1 - \cos 2\,\alpha}{2}, \operatorname{tg}^2\,\alpha = \frac{1 - \cos 2\,\alpha}{1 + \cos 2\,\alpha},
\cos^2\alpha = \frac{1 + \cos 2\,\alpha}{2}, \operatorname{ctg}^2\,\alpha = \frac{1 + \cos 2\,\alpha}{1 - \cos 2\,\alpha},
\sin^3\alpha = \frac{3\sin\alpha - \sin 3\,\alpha}{4}, \operatorname{tg}^3\,\alpha = \frac{3\sin\alpha - \sin 3\,\alpha}{3\cos\alpha + \cos 3\,\alpha},
\cos^3\alpha = \frac{3\cos\alpha + \cos 3\,\alpha}{4}, \operatorname{ctg}^3\,\alpha = \frac{3\cos\alpha + \cos 3\,\alpha}{3\sin\alpha - \sin 3\,\alpha},
\sin^4\alpha = \frac{\cos 4\alpha - 4\cos 2\,\alpha + 3}{8}, \operatorname{tg}^4\,\alpha = \frac{\cos 4\alpha - 4\cos 2\,\alpha + 3}{\cos 4\alpha + 4\cos 2\,\alpha + 3},
\cos^4\alpha = \frac{\cos 4\alpha + 4\cos 2\,\alpha + 3}{8}, \operatorname{ctg}^4\,\alpha = \frac{\cos 4\alpha + 4\cos 2\,\alpha + 3}{\cos 4\alpha - 4\cos 2\,\alpha + 3}.

Summas[izmainīt šo sadaļu | labot pirmkodu]

 \sin \alpha \pm \sin \beta = 2 \sin \frac{\alpha \pm \beta}{2} \cos \frac{\alpha \mp \beta}{2}
 \cos \alpha + \cos \beta = 2 \cos \frac{\alpha+\beta}{2} \cos \frac{\alpha-\beta}{2}
 \cos \alpha - \cos \beta = - 2 \sin \frac{\alpha+\beta}{2} \sin \frac{\alpha-\beta}{2}
 \operatorname{tg} \alpha \pm \operatorname{tg} \beta = \frac{\sin (\alpha \pm \beta)}{\cos \alpha \cos \beta}
 1 \pm \sin {2 \alpha} = (\sin \alpha \pm \cos \alpha)^2 .

Funkcijām ar argumentu x ir vienādojums:

A sin \ x + B cos \ x = \sqrt{A^2 + B^2}\sin( x + \phi ),

kur leņķi \phi atrod pēc formulas:

sin \phi =  \frac{B}{\sqrt{A^2 + B^2}}, cos \phi =  \frac{A}{\sqrt{A^2 + B^2}}.

Tangensa vienādības[izmainīt šo sadaļu | labot pirmkodu]

Jebkuru trigonometrisko funkciju var izteikt kā pusleņķa tangensu.

\sin x = \frac{\sin x}{1} = \frac{2\sin \frac{x}{2}\cos \frac{x}{2}}{\sin^2 \frac{x}{2} + \cos^2 \frac{x}{2}} =\frac{2\operatorname{tg} \frac{x}{2}}{1 + \operatorname{tg}^2 \frac{x}{2}}

\cos x = \frac{\cos x}{1} = \frac{\cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}}{\cos^2 \frac{x}{2} + \sin^2 \frac{x}{2}} =\frac{1 - \operatorname{tg}^2 \frac{x}{2}}{1 + \operatorname{tg}^2 \frac{x}{2}}

\operatorname{tg}~x = \frac{\sin x}{\cos x} = \frac{2\operatorname{tg} \frac{x}{2}}{1 - \operatorname{tg}^2 \frac{x}{2}}

\operatorname{ctg}~x = \frac{\cos x}{\sin x} = \frac{1 - \operatorname{tg}^2 \frac{x}{2}}{2\operatorname{tg} \frac{x}{2}}

\sec x = \frac{1}{\cos x} = \frac{1 + \operatorname{tg}^2 \frac{x}{2}}{1 - \operatorname{tg}^2 \frac{x}{2}}

\operatorname{cosec}~x = \frac{1}{\sin x} = \frac{1 + \operatorname{tg}^2 \frac{x}{2}} {2\operatorname{tg} \frac{x}{2}}

Trigonometrisko funkciju skaitlisko vērtību tabula[izmainīt šo sadaļu | labot pirmkodu]

θ grādos θ radiānos sin θ cos θ tan θ
0 0 0.0 1.0 0.0
1 0.017453293 0.01745240 0.9998477 0.017455065
2 0.034906585 0.034899497 0.99939083 0.034920769
3 0.052359878 0.052335956 0.99862953 0.052407779
4 0.06981317 0.069756474 0.99756405 0.069926812
5 0.087266463 0.087155743 0.9961947 0.087488664
6 0.10471976 0.10452846 0.9945219 0.10510424
7 0.12217305 0.12186934 0.99254615 0.12278456
8 0.13962634 0.1391731 0.99026807 0.14054083
9 0.15707963 0.15643447 0.98768834 0.15838444
10 0.17453293 0.17364818 0.98480775 0.17632698
11 0.19198622 0.190809 0.98162718 0.19438031
12 0.20943951 0.20791169 0.9781476 0.21255656
13 0.2268928 0.22495105 0.97437006 0.23086819
14 0.2443461 0.2419219 0.97029573 0.249328
15 0.26179939 0.25881905 0.96592583 0.26794919
16 0.27925268 0.27563736 0.9612617 0.28674539
17 0.29670597 0.2923717 0.95630476 0.30573068
18 0.31415927 0.30901699 0.95105652 0.3249197
19 0.33161256 0.32556815 0.94551858 0.34432761
20 0.34906585 0.34202014 0.93969262 0.36397023
21 0.36651914 0.35836795 0.93358043 0.38386404
22 0.38397244 0.37460659 0.92718385 0.40402623
23 0.40142573 0.39073113 0.92050485 0.42447482
24 0.41887902 0.40673664 0.91354546 0.44522869
25 0.43633231 0.42261826 0.90630779 0.46630766
26 0.45378561 0.43837115 0.89879405 0.48773259
27 0.4712389 0.4539905 0.89100652 0.50952545
28 0.48869219 0.46947156 0.88294759 0.53170943
29 0.50614548 0.48480962 0.87461971 0.55430905
30 0.52359878 0.5 0.8660254 0.57735027
31 0.54105207 0.51503807 0.8571673 0.60086062
32 0.55850536 0.52991926 0.8480481 0.62486935
33 0.57595865 0.54463904 0.83867057 0.64940759
34 0.59341195 0.5591929 0.82903757 0.67450852
35 0.61086524 0.57357644 0.81915204 0.70020754
36 0.62831853 0.58778525 0.80901699 0.72654253
37 0.64577182 0.60181502 0.79863551 0.75355405
38 0.66322512 0.61566148 0.78801075 0.78128563
39 0.68067841 0.62932039 0.77714596 0.80978403
40 0.6981317 0.64278761 0.76604444 0.83909963
41 0.71558499 0.65605903 0.75470958 0.86928674
42 0.73303829 0.66913061 0.74314483 0.90040404
43 0.75049158 0.68199836 0.7313537 0.93251509
44 0.76794487 0.69465837 0.7193398 0.96568877
45 0.78539816 0.70710678 0.70710678 1.0
46 0.80285146 0.7193398 0.69465837 1.03553031
47 0.82030475 0.7313537 0.68199836 1.07236871
48 0.83775804 0.74314483 0.66913061 1.11061251
49 0.85521133 0.75470958 0.65605903 1.15036841
50 0.87266463 0.76604444 0.64278761 1.19175359
51 0.89011792 0.77714596 0.62932039 1.23489716
52 0.90757121 0.78801075 0.61566148 1.27994163
53 0.9250245 0.79863551 0.60181502 1.32704482
54 0.9424778 0.80901699 0.58778525 1.37638192
55 0.95993109 0.81915204 0.57357644 1.42814801
56 0.97738438 0.82903757 0.5591929 1.48256097
57 0.99483767 0.83867057 0.54463904 1.53986496
58 1.01229097 0.8480481 0.52991926 1.60033453
59 1.02974426 0.8571673 0.51503807 1.66427948
60 1.04719755 0.8660254 0.5 1.73205081
61 1.06465084 0.87461971 0.48480962 1.80404776
62 1.08210414 0.88294759 0.46947156 1.88072647
63 1.09955743 0.89100652 0.4539905 1.96261051
64 1.11701072 0.89879405 0.43837115 2.05030384
65 1.13446401 0.90630779 0.42261826 2.14450692
66 1.15191731 0.91354546 0.40673664 2.24603677
67 1.1693706 0.92050485 0.39073113 2.35585237
68 1.18682389 0.92718385 0.37460659 2.47508685
69 1.20427718 0.93358043 0.35836795 2.60508906
70 1.22173048 0.93969262 0.34202014 2.74747742
71 1.23918377 0.94551858 0.32556815 2.90421088
72 1.25663706 0.95105652 0.30901699 3.07768354
73 1.27409035 0.95630476 0.2923717 3.27085262
74 1.29154365 0.9612617 0.27563736 3.48741444
75 1.30899694 0.96592583 0.25881905 3.73205081
76 1.32645023 0.97029573 0.2419219 4.01078093
77 1.34390352 0.97437006 0.22495105 4.33147587
78 1.36135682 0.9781476 0.20791169 4.70463011
79 1.37881011 0.98162718 0.190809 5.14455402
80 1.3962634 0.98480775 0.17364818 5.67128182
81 1.41371669 0.98768834 0.15643447 6.31375151
82 1.43116999 0.99026807 0.1391731 7.11536972
83 1.44862328 0.99254615 0.12186934 8.14434643
84 1.46607657 0.9945219 0.10452846 9.51436445
85 1.48352986 0.9961947 0.087155743 11.4300523
86 1.50098316 0.99756405 0.069756474 14.3006663
87 1.51843645 0.99862953 0.052335956 19.0811367
88 1.53588974 0.99939083 0.034899497 28.6362533
89 1.55334303 0.9998477 0.01745240 57.2899616
90 1.57079633 1.0 0.0

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