TRIGONOMETRIJSKE FUNKCIJE OŠTROG UGLA:
[inlmath]\sin\alpha=\frac{a}{c}[/inlmath] – sinus – odnos naspramne katete i hipotenuze
[inlmath]\cos\alpha=\frac{b}{c}[/inlmath] – kosinus – odnos nalegle katete i hipotenuze
[inlmath]\text{tg }\alpha=\frac{a}{b}[/inlmath] – tangens – odnos naspramne i nalegle katete
[inlmath]\text{ctg }\alpha=\frac{b}{a}\quad\left(=\frac{1}{\text{tg }\alpha}\right)[/inlmath] – kotangens – odnos nalegle i naspramne katete
[inlmath]\sec\alpha=\frac{c}{b}\quad\left(=\frac{1}{\cos\alpha}\right)[/inlmath] – sekans – odnos hipotenuze i nalegle katete
[inlmath]\csc\alpha=\frac{c}{a}\quad\left(=\frac{1}{\sin\alpha}\right)[/inlmath] – kosekans – odnos hipotenuze i naspramne katete
UGLOVI I KVADRANTI NA TRIGONOMETRIJSKOJ KRUŽNICI:
[inlmath]\alpha\in\left(2k\pi,\;\frac{\pi}{2}+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]I[/inlmath] kvadrantu
[inlmath]\alpha\in\left(\frac{\pi}{2}+2k\pi,\;\pi+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]II[/inlmath] kvadrantu
[inlmath]\alpha\in\left(\pi+2k\pi,\;\frac{3\pi}{2}+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]III[/inlmath] kvadrantu
[inlmath]\alpha\in\left(\frac{3\pi}{2}+2k\pi,\;2\pi+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]IV[/inlmath] kvadrantu
VREDNOSTI TRIGONOMETRIJSKIH FUNKCIJA KARAKTERISTIČNIH UGLOVA:
SVOĐENJE TRIGONOMETRIJSKIH FUNKCIJA BILO KOG UGLA NA TRIGONOMETRIJSKE FUNKCIJE OŠTROG UGLA:
[dispmath]\begin{array}{|c|}\hline
-\alpha\\ \hline
\sin\left(-\alpha\right)=-\sin\alpha\\ \hline
\cos\left(-\alpha\right)=\cos\alpha\\ \hline
\text{tg}\left(-\alpha\right)=-\text{tg }\alpha\\ \hline
\text{ctg}\left(-\alpha\right)=-\text{ctg }\alpha\\ \hline
\end{array}[/dispmath][dispmath]\begin{array}{|c|c|}\hline
\frac{\pi}{2}-\alpha & \frac{\pi}{2}+\alpha\\ \hline
\sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha & \sin\left(\frac{\pi}{2}+\alpha\right)=\cos\alpha\\ \hline
\cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha & \cos\left(\frac{\pi}{2}+\alpha\right)=-\sin\alpha\\ \hline
\text{tg}\left(\frac{\pi}{2}-\alpha\right)=\text{ctg }\alpha & \text{tg}\left(\frac{\pi}{2}+\alpha\right)=-\text{ctg }\alpha\\ \hline
\text{ctg}\left(\frac{\pi}{2}-\alpha\right)=\text{tg }\alpha & \text{ctg}\left(\frac{\pi}{2}+\alpha\right)=-\text{tg }\alpha\\ \hline
\end{array}[/dispmath][dispmath]\begin{array}{|c|c|}\hline
\pi-\alpha & -\frac{\pi}{2}+\alpha\\ \hline
\sin\left(\pi-\alpha\right)=\sin\alpha & \sin\left(-\frac{\pi}{2}+\alpha\right)=-\cos\alpha\\ \hline
\cos\left(\pi-\alpha\right)=-\cos\alpha & \cos\left(-\frac{\pi}{2}+\alpha\right)=\sin\alpha\\ \hline
\text{tg}\left(\pi-\alpha\right)=-\text{tg }\alpha & \text{tg}\left(-\frac{\pi}{2}+\alpha\right)=-\text{ctg }\alpha\\ \hline
\text{ctg}\left(\pi-\alpha\right)=-\text{ctg }\alpha & \text{ctg}\left(-\frac{\pi}{2}+\alpha\right)=-\text{tg }\alpha\\ \hline
\end{array}[/dispmath][dispmath]\begin{array}{|c|c|}\hline
-\frac{\pi}{2}-\alpha & -\pi+\alpha\\ \hline
\sin\left(-\frac{\pi}{2}-\alpha\right)=-\cos\alpha & \sin\left(-\pi+\alpha\right)=-\sin\alpha\\ \hline
\cos\left(-\frac{\pi}{2}-\alpha\right)=-\sin\alpha & \cos\left(-\pi+\alpha\right)=-\cos\alpha\\ \hline
\text{tg}\left(-\frac{\pi}{2}-\alpha\right)=\text{ctg }\alpha & \text{tg}\left(-\pi+\alpha\right)=\text{tg }\alpha\\ \hline
\text{ctg}\left(-\frac{\pi}{2}-\alpha\right)=\text{tg }\alpha & \text{ctg}\left(-\pi+\alpha\right)=\text{ctg }\alpha\\ \hline
\end{array}[/dispmath]
OSOBINE TRIGONOMETRIJSKIH FUNKCIJA:
[inlmath]-1\le\sin\alpha\le 1[/inlmath] – ograničenost sinusa
[inlmath]-1\le\cos\alpha\le 1[/inlmath] – ograničenost kosinusa
[inlmath]\sin\left(-\alpha\right)=-\sin\alpha[/inlmath] – neparnost sinusa
[inlmath]\cos\left(-\alpha\right)=\cos\alpha[/inlmath] – parnost kosinusa
[inlmath]\text{tg}\left(-\alpha\right)=-\text{tg }\alpha[/inlmath] – neparnost tangensa
[inlmath]\text{ctg}\left(-\alpha\right)=-\text{ctg }\alpha[/inlmath] – neparnost kotangensa
[inlmath]\sin\alpha=\sin\left(\alpha+2k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost sinusa s periodom [inlmath]2\pi[/inlmath]
[inlmath]\cos\alpha=\cos\left(\alpha+2k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost kosinusa s periodom [inlmath]2\pi[/inlmath]
[inlmath]\text{tg }\alpha=\text{tg}\left(\alpha+k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost tangensa s periodom [inlmath]\pi[/inlmath]
[inlmath]\text{ctg }\alpha=\text{ctg}\left(\alpha+k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost kotangensa s periodom [inlmath]\pi[/inlmath]
OSNOVNI TRIGONOMETRIJSKI IDENTITETI:
Veza sinusa i kosinusa:
[dispmath]\sin^2\alpha+\cos^2\alpha=1[/dispmath]
Veza sinusa i tangensa:
[dispmath]\sin^2\alpha=\frac{\text{tg}^2\alpha}{1+\text{tg}^2\alpha}[/dispmath]
Veza kosinusa i tangensa:
[dispmath]\cos^2\alpha=\frac{1}{1+\text{tg}^2\alpha}[/dispmath]
ADICIONE FORMULE:
Adiciona formula za sinus:
[dispmath]\sin\left(\alpha\pm\beta\right)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta[/dispmath]
Adiciona formula za kosinus:
[dispmath]\cos\left(\alpha\pm\beta\right)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta[/dispmath]
Adiciona formula za tangens:
[dispmath]\text{tg}\left(\alpha\pm\beta\right)=\frac{\text{tg }\alpha\pm\text{tg }\beta}{1\mp\text{tg }\alpha\text{ tg }\beta}[/dispmath]
Adiciona formula za kotangens:
[dispmath]\text{ctg}\left(\alpha\pm\beta\right)=\frac{\text{ctg }\alpha\text{ ctg }\beta\mp1}{\text{ctg }\beta\pm\text{ctg }\alpha}[/dispmath]
TRIGONOMETRIJSKE FORMULE DVOSTRUKOG UGLA:
[dispmath]\sin2\alpha=2\sin\alpha\cos\alpha[/dispmath][dispmath]\cos2\alpha=\cos^2\alpha-\sin^2\alpha[/dispmath][dispmath]\text{tg }2\alpha=\frac{2\text{ tg }\alpha}{1-\text{tg}^2\alpha}[/dispmath][dispmath]\text{ctg }2\alpha=\frac{\text{ctg}^2\alpha-1}{2\text{ ctg }\alpha}[/dispmath]
TRIGONOMETRIJSKE FORMULE POLOVINE UGLA:
[dispmath]\sin^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{2}[/dispmath][dispmath]\cos^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{2}[/dispmath][dispmath]\text{tg}^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{1+\cos\alpha}[/dispmath][dispmath]\text{ctg}^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{1-\cos\alpha}[/dispmath]
TRANSFORMACIJA ZBIRA I RAZLIKE TRIGONOMETRIJSKIH FUNKCIJA U PROIZVOD:
[dispmath]\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}[/dispmath][dispmath]\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}[/dispmath][dispmath]\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}[/dispmath][dispmath]\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}[/dispmath]
TRANSFORMACIJA PROIZVODA TRIGONOMETRIJSKIH FUNKCIJA U ZBIR I RAZLIKU:
[dispmath]\sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)][/dispmath][dispmath]\cos\alpha\sin\beta=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha-\beta)][/dispmath][dispmath]\cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)][/dispmath][dispmath]\sin\alpha\sin\beta=-\frac{1}{2}[\cos(\alpha+\beta)-\cos(\alpha-\beta)][/dispmath]
OSTALI TRIGONOMETRIJSKI IDENTITETI:
[dispmath]\cos2\alpha=2\cos^2\alpha-1[/dispmath][dispmath]\sin\alpha=\frac{2\text{ tg }\frac{a}{2}}{1+\text{tg}^2\frac{a}{2}}[/dispmath][dispmath]\cos\alpha=\frac{1-\text{tg}^2\frac{a}{2}}{1+\text{tg}^2\frac{a}{2}}[/dispmath][dispmath]\text{tg }\frac{a}{2}=\frac{1-\cos a}{\sin a}[/dispmath]
SINUSNA I KOSINUSNA TEOREMA:
SINUSNA TEOREMA:
[dispmath]\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R[/dispmath]
KOSINUSNA TEOREMA:
[dispmath]a^2=b^2+c^2-2bc\cos\alpha\\
b^2=a^2+c^2-2ac\cos\beta\\
c^2=a^2+b^2-2ab\cos\gamma[/dispmath]
[inlmath]a,b,c[/inlmath] – stranice trougla
[inlmath]\alpha,\beta,\gamma[/inlmath] – uglovi naspram stranica [inlmath]a,b,c[/inlmath], respektivno
[inlmath]R[/inlmath] – poluprečnik kružnice opisane oko trougla
OČITAVANJE VREDNOSTI TRIGONOMETRIJSKIH FUNKCIJA SA TRIGONOMETRIJSKE KRUŽNICE:
OČITAVANJE PREDZNAKA TRIGONOMETRIJSKIH FUNKCIJA SA TRIGONOMETRIJSKE KRUŽNICE:
GRAFICI TRIGONOMETRIJSKIH FUNKCIJA:
[inlmath]\sin\alpha=\frac{a}{c}[/inlmath] – sinus – odnos naspramne katete i hipotenuze
[inlmath]\cos\alpha=\frac{b}{c}[/inlmath] – kosinus – odnos nalegle katete i hipotenuze
[inlmath]\text{tg }\alpha=\frac{a}{b}[/inlmath] – tangens – odnos naspramne i nalegle katete
[inlmath]\text{ctg }\alpha=\frac{b}{a}\quad\left(=\frac{1}{\text{tg }\alpha}\right)[/inlmath] – kotangens – odnos nalegle i naspramne katete
[inlmath]\sec\alpha=\frac{c}{b}\quad\left(=\frac{1}{\cos\alpha}\right)[/inlmath] – sekans – odnos hipotenuze i nalegle katete
[inlmath]\csc\alpha=\frac{c}{a}\quad\left(=\frac{1}{\sin\alpha}\right)[/inlmath] – kosekans – odnos hipotenuze i naspramne katete
UGLOVI I KVADRANTI NA TRIGONOMETRIJSKOJ KRUŽNICI:
[inlmath]\alpha\in\left(2k\pi,\;\frac{\pi}{2}+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]I[/inlmath] kvadrantu
[inlmath]\alpha\in\left(\frac{\pi}{2}+2k\pi,\;\pi+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]II[/inlmath] kvadrantu
[inlmath]\alpha\in\left(\pi+2k\pi,\;\frac{3\pi}{2}+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]III[/inlmath] kvadrantu
[inlmath]\alpha\in\left(\frac{3\pi}{2}+2k\pi,\;2\pi+2k\pi\right)\quad\Longrightarrow\quad\alpha[/inlmath] pripada [inlmath]IV[/inlmath] kvadrantu
VREDNOSTI TRIGONOMETRIJSKIH FUNKCIJA KARAKTERISTIČNIH UGLOVA:
[dispmath]\alpha[/dispmath] | [dispmath]0[/dispmath] | [dispmath]\frac{\pi}{6}[/dispmath] | [dispmath]\frac{\pi}{4}[/dispmath] | [dispmath]\frac{\pi}{3}[/dispmath] | [dispmath]\frac{\pi}{2}[/dispmath] |
[dispmath]\sin\alpha[/dispmath] | [dispmath]0[/dispmath] | [dispmath]\frac{1}{2}[/dispmath] | [dispmath]\;\;\frac{\sqrt 2}{2}\;\;[/dispmath] | [dispmath]\;\;\frac{\sqrt 3}{2}\;\;[/dispmath] | [dispmath]1[/dispmath] |
[dispmath]\cos\alpha[/dispmath] | [dispmath]1[/dispmath] | [dispmath]\;\;\frac{\sqrt 3}{2}\;\;[/dispmath] | [dispmath]\frac{\sqrt 2}{2}[/dispmath] | [dispmath]\frac{1}{2}[/dispmath] | [dispmath]0[/dispmath] |
[dispmath]\text{tg }\alpha[/dispmath] | [dispmath]0[/dispmath] | [dispmath]\frac{\sqrt 3}{3}[/dispmath] | [dispmath]1[/dispmath] | [dispmath]\sqrt 3[/dispmath] | [dispmath]\;\;\infty\;\;[/dispmath] |
[dispmath]\;\;\text{ctg }\alpha\;\;[/dispmath] | [dispmath]\;\;\infty\;\;[/dispmath] | [dispmath]\sqrt 3[/dispmath] | [dispmath]1[/dispmath] | [dispmath]\frac{\sqrt 3}{3}[/dispmath] | [dispmath]0[/dispmath] |
SVOĐENJE TRIGONOMETRIJSKIH FUNKCIJA BILO KOG UGLA NA TRIGONOMETRIJSKE FUNKCIJE OŠTROG UGLA:
[dispmath]\begin{array}{|c|}\hline
-\alpha\\ \hline
\sin\left(-\alpha\right)=-\sin\alpha\\ \hline
\cos\left(-\alpha\right)=\cos\alpha\\ \hline
\text{tg}\left(-\alpha\right)=-\text{tg }\alpha\\ \hline
\text{ctg}\left(-\alpha\right)=-\text{ctg }\alpha\\ \hline
\end{array}[/dispmath][dispmath]\begin{array}{|c|c|}\hline
\frac{\pi}{2}-\alpha & \frac{\pi}{2}+\alpha\\ \hline
\sin\left(\frac{\pi}{2}-\alpha\right)=\cos\alpha & \sin\left(\frac{\pi}{2}+\alpha\right)=\cos\alpha\\ \hline
\cos\left(\frac{\pi}{2}-\alpha\right)=\sin\alpha & \cos\left(\frac{\pi}{2}+\alpha\right)=-\sin\alpha\\ \hline
\text{tg}\left(\frac{\pi}{2}-\alpha\right)=\text{ctg }\alpha & \text{tg}\left(\frac{\pi}{2}+\alpha\right)=-\text{ctg }\alpha\\ \hline
\text{ctg}\left(\frac{\pi}{2}-\alpha\right)=\text{tg }\alpha & \text{ctg}\left(\frac{\pi}{2}+\alpha\right)=-\text{tg }\alpha\\ \hline
\end{array}[/dispmath][dispmath]\begin{array}{|c|c|}\hline
\pi-\alpha & -\frac{\pi}{2}+\alpha\\ \hline
\sin\left(\pi-\alpha\right)=\sin\alpha & \sin\left(-\frac{\pi}{2}+\alpha\right)=-\cos\alpha\\ \hline
\cos\left(\pi-\alpha\right)=-\cos\alpha & \cos\left(-\frac{\pi}{2}+\alpha\right)=\sin\alpha\\ \hline
\text{tg}\left(\pi-\alpha\right)=-\text{tg }\alpha & \text{tg}\left(-\frac{\pi}{2}+\alpha\right)=-\text{ctg }\alpha\\ \hline
\text{ctg}\left(\pi-\alpha\right)=-\text{ctg }\alpha & \text{ctg}\left(-\frac{\pi}{2}+\alpha\right)=-\text{tg }\alpha\\ \hline
\end{array}[/dispmath][dispmath]\begin{array}{|c|c|}\hline
-\frac{\pi}{2}-\alpha & -\pi+\alpha\\ \hline
\sin\left(-\frac{\pi}{2}-\alpha\right)=-\cos\alpha & \sin\left(-\pi+\alpha\right)=-\sin\alpha\\ \hline
\cos\left(-\frac{\pi}{2}-\alpha\right)=-\sin\alpha & \cos\left(-\pi+\alpha\right)=-\cos\alpha\\ \hline
\text{tg}\left(-\frac{\pi}{2}-\alpha\right)=\text{ctg }\alpha & \text{tg}\left(-\pi+\alpha\right)=\text{tg }\alpha\\ \hline
\text{ctg}\left(-\frac{\pi}{2}-\alpha\right)=\text{tg }\alpha & \text{ctg}\left(-\pi+\alpha\right)=\text{ctg }\alpha\\ \hline
\end{array}[/dispmath]
OSOBINE TRIGONOMETRIJSKIH FUNKCIJA:
[inlmath]-1\le\sin\alpha\le 1[/inlmath] – ograničenost sinusa
[inlmath]-1\le\cos\alpha\le 1[/inlmath] – ograničenost kosinusa
[inlmath]\sin\left(-\alpha\right)=-\sin\alpha[/inlmath] – neparnost sinusa
[inlmath]\cos\left(-\alpha\right)=\cos\alpha[/inlmath] – parnost kosinusa
[inlmath]\text{tg}\left(-\alpha\right)=-\text{tg }\alpha[/inlmath] – neparnost tangensa
[inlmath]\text{ctg}\left(-\alpha\right)=-\text{ctg }\alpha[/inlmath] – neparnost kotangensa
[inlmath]\sin\alpha=\sin\left(\alpha+2k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost sinusa s periodom [inlmath]2\pi[/inlmath]
[inlmath]\cos\alpha=\cos\left(\alpha+2k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost kosinusa s periodom [inlmath]2\pi[/inlmath]
[inlmath]\text{tg }\alpha=\text{tg}\left(\alpha+k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost tangensa s periodom [inlmath]\pi[/inlmath]
[inlmath]\text{ctg }\alpha=\text{ctg}\left(\alpha+k\pi\right),\;k\in\mathbb{Z}[/inlmath] – periodičnost kotangensa s periodom [inlmath]\pi[/inlmath]
OSNOVNI TRIGONOMETRIJSKI IDENTITETI:
Veza sinusa i kosinusa:
[dispmath]\sin^2\alpha+\cos^2\alpha=1[/dispmath]
Veza sinusa i tangensa:
[dispmath]\sin^2\alpha=\frac{\text{tg}^2\alpha}{1+\text{tg}^2\alpha}[/dispmath]
Veza kosinusa i tangensa:
[dispmath]\cos^2\alpha=\frac{1}{1+\text{tg}^2\alpha}[/dispmath]
ADICIONE FORMULE:
Adiciona formula za sinus:
[dispmath]\sin\left(\alpha\pm\beta\right)=\sin\alpha\cos\beta\pm\cos\alpha\sin\beta[/dispmath]
Adiciona formula za kosinus:
[dispmath]\cos\left(\alpha\pm\beta\right)=\cos\alpha\cos\beta\mp\sin\alpha\sin\beta[/dispmath]
Adiciona formula za tangens:
[dispmath]\text{tg}\left(\alpha\pm\beta\right)=\frac{\text{tg }\alpha\pm\text{tg }\beta}{1\mp\text{tg }\alpha\text{ tg }\beta}[/dispmath]
Adiciona formula za kotangens:
[dispmath]\text{ctg}\left(\alpha\pm\beta\right)=\frac{\text{ctg }\alpha\text{ ctg }\beta\mp1}{\text{ctg }\beta\pm\text{ctg }\alpha}[/dispmath]
TRIGONOMETRIJSKE FORMULE DVOSTRUKOG UGLA:
[dispmath]\sin2\alpha=2\sin\alpha\cos\alpha[/dispmath][dispmath]\cos2\alpha=\cos^2\alpha-\sin^2\alpha[/dispmath][dispmath]\text{tg }2\alpha=\frac{2\text{ tg }\alpha}{1-\text{tg}^2\alpha}[/dispmath][dispmath]\text{ctg }2\alpha=\frac{\text{ctg}^2\alpha-1}{2\text{ ctg }\alpha}[/dispmath]
TRIGONOMETRIJSKE FORMULE POLOVINE UGLA:
[dispmath]\sin^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{2}[/dispmath][dispmath]\cos^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{2}[/dispmath][dispmath]\text{tg}^2\frac{\alpha}{2}=\frac{1-\cos\alpha}{1+\cos\alpha}[/dispmath][dispmath]\text{ctg}^2\frac{\alpha}{2}=\frac{1+\cos\alpha}{1-\cos\alpha}[/dispmath]
TRANSFORMACIJA ZBIRA I RAZLIKE TRIGONOMETRIJSKIH FUNKCIJA U PROIZVOD:
[dispmath]\sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}[/dispmath][dispmath]\sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}[/dispmath][dispmath]\cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}[/dispmath][dispmath]\cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}[/dispmath]
TRANSFORMACIJA PROIZVODA TRIGONOMETRIJSKIH FUNKCIJA U ZBIR I RAZLIKU:
[dispmath]\sin\alpha\cos\beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha-\beta)][/dispmath][dispmath]\cos\alpha\sin\beta=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha-\beta)][/dispmath][dispmath]\cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha-\beta)][/dispmath][dispmath]\sin\alpha\sin\beta=-\frac{1}{2}[\cos(\alpha+\beta)-\cos(\alpha-\beta)][/dispmath]
OSTALI TRIGONOMETRIJSKI IDENTITETI:
[dispmath]\cos2\alpha=2\cos^2\alpha-1[/dispmath][dispmath]\sin\alpha=\frac{2\text{ tg }\frac{a}{2}}{1+\text{tg}^2\frac{a}{2}}[/dispmath][dispmath]\cos\alpha=\frac{1-\text{tg}^2\frac{a}{2}}{1+\text{tg}^2\frac{a}{2}}[/dispmath][dispmath]\text{tg }\frac{a}{2}=\frac{1-\cos a}{\sin a}[/dispmath]
SINUSNA I KOSINUSNA TEOREMA:
SINUSNA TEOREMA:
[dispmath]\frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}=2R[/dispmath]
KOSINUSNA TEOREMA:
[dispmath]a^2=b^2+c^2-2bc\cos\alpha\\
b^2=a^2+c^2-2ac\cos\beta\\
c^2=a^2+b^2-2ab\cos\gamma[/dispmath]
[inlmath]a,b,c[/inlmath] – stranice trougla
[inlmath]\alpha,\beta,\gamma[/inlmath] – uglovi naspram stranica [inlmath]a,b,c[/inlmath], respektivno
[inlmath]R[/inlmath] – poluprečnik kružnice opisane oko trougla
OČITAVANJE VREDNOSTI TRIGONOMETRIJSKIH FUNKCIJA SA TRIGONOMETRIJSKE KRUŽNICE:
OČITAVANJE PREDZNAKA TRIGONOMETRIJSKIH FUNKCIJA SA TRIGONOMETRIJSKE KRUŽNICE:
GRAFICI TRIGONOMETRIJSKIH FUNKCIJA: