Trigonometrija - sin, cos, tg, ctg

Uzmimo x-osu i y-osu koordinatnog sistema i O za koordinatni početak. Kružnicu sa centrom u O poluprečnika = 1 zovemo trigonometrijska kružnica ili jedinična kružnica.
trigonometrijska kružnica
Ako je P tačka kružnice i t ugao između PO i x onda:
  • x-koordinatu tačke P zovemo kosinus ugla t. Pišemo: cos(t);
  • y-koordinatu tačke P zovemo sinus ugla t. Pišemo: sin(t);
  • broj sin(t)/cos(t) zovemo tangens ugla t. Pišemo: tg(t);
  • broj cos(t)/sin(t) zovemo kotangens ugla t. Pišemo: ctg(t).

Sinusna funkcija

sin : R -> R
Sve trigonometrijske funkcije su periodične. Period sinusne funkcije je .
Kodomen: [-1,1].

grafik sinusne funkcije

Kosinusna funkcija

cos : R -> R
Period kosinusne funkcije je .
Kodomen: [-1,1].

grafik kosinusne funkcije

Tangensna funkcija

tg : R -> R
Kodomen: R. Period je π a funkcija nije definisana za x = (π/2) + kπ, k=0,1,2,...

grafik tangensne funkcije

Kotangensna funkcija

ctg : R -> R
Kodomen: R. Period je π a funkcija nije definisana za x = kπ, k=0,1,2,...

grafik kotangensne funkcije

Vrednosti sin, cos, tg, ctg za uglove 0°, 30°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°

$\alpha^o$ $0^o$ $30^o$ $45^o$ $60^o$ $90^o$ $120^o$ $135^o$ $150^o$ $180^o$ $210^o$ $225^o$ $240^o$ $270^o$ $300^o$ $315^o$ $330^o$ $360^o$
$\alpha rad$ $0$ $\frac{\pi}{6}$ $\frac{\pi}{4}$ $\frac{\pi}{3}$ $\frac{\pi}{2}$ $\frac{2\pi}{3}$ $\frac{3\pi}{4}$ $\frac{5\pi}{6}$ $\pi$ $\frac{7\pi}{6}$ $\frac{5\pi}{4}$ $\frac{4\pi}{3}$ $\frac{3\pi}{2}$ $\frac{5\pi}{3}$ $\frac{7\pi}{4}$ $\frac{11\pi}{6}$ $2\pi$
$sin\alpha$ $0$ $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{3}}{2}$ $-1$ $-\frac{\sqrt{3}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{1}{2}$ $0$
$cos\alpha$ $1$ $\frac{\sqrt{3}}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{1}{2}$ $0$ $-\frac{1}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{\sqrt{3}}{2}$ $-1$ $-\frac{\sqrt{3}}{2}$ $-\frac{\sqrt{2}}{2}$ $-\frac{1}{2}$ $0$ $\frac{1}{2}$ $\frac{\sqrt{2}}{2}$ $\frac{\sqrt{3}}{2}$ $1$
$tg\alpha$ $0$ $\frac{\sqrt{3}}{3}$ $1$ $\sqrt{3}$ $-$ $-\sqrt{3}$ $-1$ $-\frac{\sqrt{3}}{3}$ $0$ $\frac{\sqrt{3}}{3}$ $1$ $\sqrt{3}$ $-$ $-\sqrt{3}$ $-1$ $-\frac{\sqrt{3}}{3}$ $0$
$ctg\alpha$ $-$ $\sqrt{3}$ $1$ $\frac{\sqrt{3}}{3}$ $0$ $-\frac{\sqrt{3}}{3}$ $-1$ $-\sqrt{3}$ $-$ $\sqrt{3}$ $1$ $\frac{\sqrt{3}}{3}$ $0$ $-\frac{\sqrt{3}}{3}$ $-1$ $-\sqrt{3}$ $-$

Najlakši način za pamćenje vrednosti funkcija sin i cos za uglove 0°, 30°, 60°, 90°:
sin([0, 30, 45, 60, 90]) = cos([90, 60, 45, 30, 0]) = sqrt([0, 1, 2, 3, 4]/4)

Trigonometrijski identiteti

Uglu od t radiana odgovara tačno jedna tačka P(cos(t),sin(t)) na jediničnoj kružnici. Udaljenost [OP] = 1. Izračunavanje rastojanja tačke P za svako t:

cos2(t) + sin2(t) = 1

Ako je t + t' = 180° onda je:

  • sin(t) = sin(t')
  • cos(t) = -cos(t')
  • tg(t) = -tg(t')
  • ctg(t) = -ctg(t')

Ako je t + t' = 90° onda je:

  • sin(t) = cos(t')
  • cos(t) = sin(t')
  • tg(t) = ctg(t')
  • ctg(t) = tg(t')

$-\alpha$ $90^\circ - \alpha$ $90^\circ + \alpha$ $180^\circ - \alpha$
$\textrm{ sin }$ $-\textrm{ sin }\alpha$ $\textrm{ cos }\alpha$ $\textrm{ cos } \alpha$ $\textrm{ sin }\alpha$
$\textrm{ cos }$ $\textrm{ cos }\alpha$ $\textrm{ sin }\alpha$ $-\textrm{ sin} \alpha$ $-\textrm{ cos }\alpha$
$\textrm{ tg }$ $-\textrm{ tg }\alpha$ $\textrm{ ctg }\alpha$ $-\textrm{ ctg } \alpha$ $-\textrm{ tg }\alpha$
$\textrm{ ctg }$ $-\textrm{ ctg }\alpha$ $\textrm{ tg }\alpha$ $-\textrm{ tg } \alpha$ $-\textrm{ ctg }\alpha$

Trigonometrijske formule

Formule polovičnog ugle

$\sin\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{2}}$
+ ako $\frac{\alpha}{2}$ leži u kvadrantu | ili ||
- ako $\frac{\alpha}{2}$ leži u kvadrantu ||| ili |V


$\cos\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{2}}$
+ ako $\frac{\alpha}{2}$ leži u kvadrantu | ili |V
- ako $\frac{\alpha}{2}$ leži u kvadrantu || ili |||


$tg\frac{\alpha}{2}=\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}$
+ ako $\frac{\alpha}{2}$ leži u kvadrantu | ili |||
- ako $\frac{\alpha}{2}$ leži u kvadrantu || ili |V


$\textrm{ ctg }\frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}$
+ ako $\frac{\alpha}{2}$ leži u kvadrantu | ili |||
- ako $\frac{\alpha}{2}$ leži u kvadrantu || ili |V


$\textrm{ tg }\frac{\alpha}{2} = \frac{\sin\alpha}{1+\cos\alpha} = \frac{1-\cos\alpha}{\sin\alpha}=\csc\alpha-\textrm{ ctg }\alpha$

$\textrm{ ctg }\frac{\alpha}{2} = \frac{\sin\alpha}{1-\cos\alpha} = \frac{1+\cos\alpha}{\sin\alpha}=\csc\alpha+\textrm{ ctg }\alpha$

Formule dvostrukog/trostrukog ugla

$\sin(2u) = 2\sin(u)\cdot \cos(u)$

$\cos(2u) = \cos^2(u) - \sin^2(u) = 2\cos^2(u) - 1 = 1 - 2\sin^2(u)$

$\textrm{ tg }(2u) = \frac{2\textrm{ tg }(u)}{1- \textrm{ tg }^2(u)}$

$\cos(2u) = \frac{1 - \textrm{ tg }^2(u)}{1 + \textrm{ tg }^2(u)}$

$\sin(2u) = \frac{2\textrm{ tg }(u)}{1 + \textrm{ tg }^2(u)}$


$\sin3\alpha = 3\sin\alpha - 4 \sin^3\alpha$

$\cos3\alpha = 4\cos^3\alpha - 3 \cos\alpha$

$\textrm{ tg }3\alpha=\frac{3\textrm{ tg }\alpha - \textrm{ tg }^3\alpha}{1-3\textrm{ tg }^2\alpha}$

$\textrm{ ctg }3\alpha=\frac{\textrm{ ctg }^3\alpha-3\textrm{ ctg }\alpha}{3\textrm{ ctg }^2\alpha-1}$


$\sin4\alpha = 4\cos^3\alpha\sin\alpha - 4\cos\alpha \sin^3\alpha$

$\cos4\alpha = \cos^4\alpha - 6\cos^2\alpha\sin^2\alpha + \sin^4\alpha$

$\textrm{ tg }4\alpha=\frac{4\textrm{ tg }\alpha - 4\textrm{ tg }^3\alpha}{1-6\textrm{ tg }^2\alpha+\textrm{ tg }^4\alpha}$

$\textrm{ ctg }4\alpha=\frac{\textrm{ ctg }^4\alpha-6\textrm{ ctg }^2\alpha+1}{4\textrm{ ctg }^3\alpha-4\textrm{ ctg }\alpha}$

Stepenovanje funkcija

$\sin^2(\alpha)=\frac{1 - \cos(2\alpha)}{2}$

$\sin^3(\alpha)=\frac{3\sin\alpha - \sin(3\alpha)}{4}$

$\sin^4(\alpha)=\frac{\cos(4\alpha) - 4\cos(2\alpha) + 3}{8}$


$\cos^2(\alpha) = \frac{1 + \cos(2\alpha)}{2}$

$\cos^3(\alpha)=\frac{3\cos\alpha + \cos(3\alpha)}{4}$

$\cos^4(\alpha)=\frac{4\cos(2\alpha) + \cos(4\alpha) + 3}{8}$

Funkcije zbira i razlike

$\sin(\alpha + \beta) = \sin(\alpha)\cdot \cos(\beta) + \cos(\alpha)\cdot \sin(\beta)$

$\sin(\alpha - \beta) = \sin(\alpha)\cdot \cos(\beta) - \cos(\alpha)\cdot \sin(\beta)$


$\cos(\alpha + \beta) = \cos(\alpha)\cdot \cos(\beta) - \sin(\alpha)\cdot \sin(\beta)$

$\cos(\alpha - \beta) = \cos(\alpha)\cdot \cos(\beta) + \sin(\alpha)\cdot \sin(\beta)$


$\textrm{ tg }(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)}=\frac{\sin(\alpha)\cdot \cos(\beta) + \cos(\alpha)\cdot \sin(\beta)}{\cos(\alpha)\cdot \cos(\beta) - \sin(\alpha)\cdot \sin(\beta)}$

$\textrm{ tg }(\alpha + \beta) = \frac{\textrm{ tg }(\alpha) + \textrm{ tg }(\beta)}{1 - \textrm{ tg }(\alpha)\cdot\textrm{ tg }(\beta)}$

$\textrm{ ctg }(\alpha \pm \beta) = \frac{\textrm{ ctg }(\beta)\textrm{ ctg }(\alpha)\mp 1}{\textrm{ ctg }(\beta)\pm cot(\alpha)}=\frac{1\mp \textrm{ tg }(\alpha)\textrm{ tg }(\beta)}{\textrm{ tg }(\alpha)\pm \textrm{ tg }(\beta)}$


$\sin(\alpha + \beta + \gamma) = \sin\alpha \cos\beta \cos\gamma + \cos\alpha \sin\beta \cos\gamma + \cos\alpha \cos\beta \sin\gamma - \sin\alpha \sin\beta \sin\gamma$

$\cos(\alpha + \beta + \gamma) = \cos\alpha \cos\beta \cos\gamma - \sin\alpha \sin\beta \cos\gamma - \sin\alpha \cos\beta \sin\gamma $
$- \sin\alpha \cos\beta \sin\gamma - \cos\alpha \sin\beta \sin\gamma$

$\textrm{ tg }(\alpha + \beta + \gamma) = \frac{\textrm{ tg }\alpha + \textrm{ tg }\beta + \textrm{ tg }\gamma - \textrm{ tg }\alpha\cdot \textrm{ tg }\beta \cdot \textrm{ tg }\gamma}{1 - \textrm{ tg }\alpha\cdot\textrm{ tg }\beta - \textrm{ tg }\beta\cdot\textrm{ tg }\gamma - \textrm{ tg }\alpha\cdot\textrm{ tg }\gamma}$

Zbir i razlika funkcija

$\textrm{ sin } \alpha + \textrm{ sin }\beta = 2 \textrm{ sin }\frac{\alpha + \beta}{2} \textrm{ cos }\frac{\alpha - \beta}{2}$

$\textrm{ sin } \alpha - \textrm{ sin }\beta = 2 \textrm{ sin }\frac{\alpha - \beta}{2} \textrm{ cos }\frac{\alpha + \beta}{2}$


$\textrm{ cos } \alpha + \textrm{ cos }\beta = 2 \textrm{ cos }\frac{\alpha + \beta}{2} \textrm{ cos }\frac{\alpha - \beta}{2}$

$\textrm{ cos } \alpha - \textrm{ cos }\beta = -2 \textrm{ sin }\frac{\alpha + \beta}{2} \textrm{ sin }\frac{\alpha - \beta}{2}$


$\textrm{ tg }\alpha + \textrm{ tg }\beta = \frac{\sin(\alpha+\beta)}{\cos\alpha\cdot\cos\beta}$

$\textrm{ tg }\alpha - \textrm{ tg }\beta = \frac{\sin(\alpha-\beta)}{\cos\alpha\cdot\cos\beta}$


$\textrm{ ctg }\alpha + \textrm{ ctg }\beta = \frac{\sin(\alpha+\beta)}{\sin\alpha\cdot\sin\beta}$

$\textrm{ ctg }\alpha - \textrm{ ctg }\beta = \frac{-\sin(\alpha-\beta)}{\sin\alpha\cdot\sin\beta}$


$\textrm{ sin }\alpha \textrm{ sin }\beta = \frac{1}{2} (\textrm{ cos }(\alpha - \beta) - \textrm{ cos }(\alpha + \beta))$

$\textrm{ cos }\alpha \textrm{ cos }\beta = \frac{1}{2} (\textrm{ cos }(\alpha - \beta) + \textrm{ cos }(\alpha + \beta))$

$\textrm{ sin }\alpha \textrm{ cos }\beta = \frac{1}{2} (\textrm{ sin }(\alpha + \beta) + \textrm{ sin }(\alpha - \beta))$


$\textrm{ tg }\alpha\textrm{ tg }\beta = \frac{\textrm{ tg }\alpha+\textrm{ tg }\beta}{\textrm{ ctg }\alpha+\textrm{ ctg }\beta}=-\frac{\textrm{ tg }\alpha-\textrm{ tg }\beta}{\textrm{ ctg }\alpha-\textrm{ ctg }\beta}$

$\textrm{ ctg }\alpha\textrm{ ctg }\beta = \frac{\textrm{ ctg }\alpha+\textrm{ ctg }\beta}{\textrm{ tg }\alpha+\textrm{ tg }\beta}$

$\textrm{ tg }\alpha\textrm{ ctg }\beta = \frac{\textrm{ tg }\alpha+\textrm{ ctg }\beta}{\textrm{ ctg }\alpha+\textrm{ tg }\beta}$


$\sin\alpha\sin\beta\sin\gamma = \frac{1}{4}\big(\sin(\alpha+\beta-\gamma)+\sin(\beta+\gamma-\alpha)+\sin(\gamma+\alpha-\beta)-\sin(\alpha+\beta+\gamma)\big)$

$\cos\alpha\cos\beta\cos\gamma = \frac{1}{4}\big(\cos(\alpha+\beta-\gamma)+\cos(\beta+\gamma-\alpha)+\cos(\gamma+\alpha-\beta)+\cos(\alpha+\beta+\gamma)\big)$

$\sin\alpha\sin\beta\cos\gamma = \frac{1}{4}\big(-\cos(\alpha+\beta-\gamma)+\cos(\beta+\gamma-\alpha)+\cos(\gamma+\alpha-\beta)-\cos(\alpha+\beta+\gamma)\big)$

$\sin\alpha\cos\beta\cos\gamma = \frac{1}{4}\big(\sin(\alpha+\beta-\gamma)-\sin(\beta+\gamma-\alpha)+\sin(\gamma+\alpha-\beta)+\sin(\alpha+\beta+\gamma)\big)$


$\sin\alpha = \frac{2\textrm{tg}\frac{\alpha}{2}}{1+\textrm{tg}^2\frac{\alpha}{2}}$

$\cos\alpha = \frac{1-\textrm{tg}^2\frac{\alpha}{2}}{1+\textrm{tg}^2\frac{\alpha}{2}}$

$\textrm{tg}\alpha = \frac{2\textrm{tg}\frac{\alpha}{2}}{1-\textrm{tg}^2\frac{\alpha}{2}}$

$\textrm{ctg}\alpha = \frac{1-\textrm{tg}^2\frac{\alpha}{2}}{2\textrm{tg}\frac{\alpha}{2}}$


$1\pm\sin\alpha=2\sin^2\big(\frac{\pi}{4}\pm \frac{\alpha}{2}\big)=2\cos^2\big(\frac{\pi}{4}\mp \frac{\alpha}{2}\big)$


$\frac{1-\sin\alpha}{1+\sin\alpha} = \textrm{ tg }^2(\frac{\pi}{4}-\frac{\alpha}{2})$

$\frac{1-\cos\alpha}{1+\cos\alpha} = \textrm{ tg }^2\frac{\alpha}{2}$

$\frac{1-\textrm{ tg }\alpha}{1+\textrm{ tg }\alpha} = \textrm{ tg }(\frac{\pi}{4}-\alpha)$

$\frac{1+\textrm{ tg }\alpha}{1-\textrm{ tg }\alpha} = \textrm{ tg }(\frac{\pi}{4}+\alpha)$

$\frac{\textrm{ ctg }\alpha + 1}{\textrm{ ctg }\alpha - 1} = \textrm{ ctg }(\frac{\pi}{4}-\alpha)$

$\textrm{ tg }\alpha + \textrm{ ctg }\alpha = \frac{2}{\sin2\alpha}$

$\textrm{ tg }\alpha - \textrm{ ctg }\alpha = -2\textrm{ ctg }2\alpha$


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