Definicije

$$\sin \alpha =\frac{a}{c}$$

$$\cos \alpha =\frac{b}{c}$$

$$\tan \alpha =\frac{a}{b}$$

$$\cot \alpha =\frac{b}{a}$$

Osnovne trigonometrijske jednakosti

Ako je \(0^{\circ}<\alpha <90^{\circ}\), onda je:

  • $$\sin ^2\alpha +\cos ^2\alpha =1$$
  • $$\tan \alpha =\frac{\sin \alpha }{\cos \alpha }=\frac{1}{\cot \alpha }$$
  • $$\cot \alpha =\frac{\cos \alpha }{\sin \alpha }=\frac{1}{\tan \alpha }$$
  • $$\tan \alpha \cdot \cot \alpha =1$$
  • $$\sin \alpha =\frac{\tan \alpha }{\sqrt{1+\tan ^2\alpha }}; \; \cos \alpha =\frac{1}{\sqrt{1+\tan ^2\alpha }}$$
  • $$1+\tan ^2\alpha =\frac{1}{\cos ^2\alpha }; \; 1+\cot ^2\alpha =\frac{1}{\sin ^2\alpha }$$

Svođenje trigonometrijskih funkcija na oštar ugao

$$90^{\circ}=\frac{\pi }{2};\; 180^{\circ}=\pi; \; 270^{\circ}=\frac{3\pi }{2};360^{\circ}=2\pi $$

\(\sin\) \(\cos\) \(\tan\) \(\cot\)
\(- \alpha\) \(-\sin \alpha\) \(+\cos \alpha\) \(-\tan \alpha\) \(-\cot \alpha\)
\(90^{\circ}-\alpha \) \(+\cos \alpha\) \(+\sin \alpha\) \(+\cot \alpha\) \(+\tan \alpha\)
\(90^{\circ}+\alpha \) \(+\cos \alpha\) \(-\sin \alpha\) \(-\cot \alpha\) \(-\tan \alpha\)
\(180^{\circ}-\alpha \) \(+\sin \alpha\) \(-\cos \alpha\) \(-\tan \alpha\) \(-\cot \alpha\)
\(180^{\circ}+\alpha \) \(-\sin \alpha\) \(-\cos \alpha\) \(+\tan \alpha\) \(+\cot \alpha\)
\(270^{\circ}-\alpha \) \(-\cos \alpha\) \(-\sin \alpha\) \(+\cot \alpha\) \(+\tan \alpha\)
\(270^{\circ}+\alpha \) \(-\cos \alpha\) \(+\sin \alpha\) \(-\cot \alpha\) \(-\tan \alpha\)
\(360^{\circ}-\alpha \)\(-\sin \alpha\)\(+\cos \alpha\)\(-\tan \alpha\)\(-\cot \alpha\)
\(360^{\circ}+\alpha \) \(+\sin \alpha\)\(+\cos \alpha\)\(+\tan \alpha\)\(-\cot \alpha\)

Vrednosti trigonometrijskih funkcija

\(\alpha\) \(0^{\circ}\) \(30^{\circ}\) \(45^{\circ}\) \(60^{\circ}\) \(90^{\circ}\)
\(\sin \alpha\) \(0\) \(\frac{1}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{\sqrt{3}}{2}\) \(1\)
\(\cos \alpha\) \(1\)\(\frac{\sqrt{3}}{2}\) \(\frac{\sqrt{2}}{2}\) \(\frac{1}{2}\) \(0\)
\(\tan \alpha\) \(0\) \(\frac{\sqrt{3}}{3}\) \(1\) \(\sqrt{3}\) \(\infty \)
\(\cot \alpha\) \(\infty \) \(\sqrt{3}\) \(1\) \(\frac{\sqrt{3}}{3}\) \(0\)

Trigonometrijske funkcije zbira i razlike dva ugla

$$\sin (\alpha \pm \beta )=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta $$

$$\cos (\alpha \pm \beta )=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta $$

$$\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta }{1\mp \tan \alpha \tan \beta }$$

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

Trigonometrijske funkcije dvostrukog i trostrukog ugla

$$\sin2\alpha =2\sin \alpha \cos \alpha$$

$$\cos 2\alpha =\cos ^2\alpha -\sin ^2\alpha =2\cos ^2-1=1-2\sin ^2\alpha $$

$$\tan 2\alpha =\frac{2\tan \alpha }{1-\tan ^2\alpha }$$

$$\cot 2\alpha =\frac{\cot ^2-1}{2\cot \alpha}$$

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

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

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

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

Trigonometrijske funkcije polovine ugla

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

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

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

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

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

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

$$\sin \alpha =\frac{2\tan \frac{\alpha }{2}}{1-\tan ^2\frac{\alpha }{2}}$$

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

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

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

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

$$1+\sin \alpha =\left ( \sin \frac{\alpha }{2}+\cos \frac{\alpha }{2} \right )^2=2\cos ^2\left ( \frac{\pi }{4}-\frac{\alpha }{2} \right )$$

$$1-\sin \alpha =\left ( \sin \frac{\alpha }{2}-\cos \frac{\alpha }{2} \right )^2=2\sin ^2\left ( \frac{\pi }{4}-\frac{\alpha }{2} \right )$$

$$\sin ^3\alpha =\frac{1}{4}(3\sin \alpha -\sin3 \alpha )$$

$$\cos ^3\alpha =\frac{1}{4}(3\cos \alpha +\cos 3\alpha )$$

Trigonometrijske funkcije polovine ugla

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

Transformacija zbira i razlike trigonometrijskih funkcija

  • $$\sin \alpha +\sin \beta =2\sin \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$$
  • $$\sin \alpha -\sin \beta =2\cos \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$
  • $$\cos \alpha +\cos \beta =2\cos \frac{\alpha +\beta }{2}\cos \frac{\alpha -\beta }{2}$$
  • $$\cos \alpha -\cos \beta =-2\sin \frac{\alpha +\beta }{2}\sin \frac{\alpha -\beta }{2}$$
  • $$\tan \alpha \pm \tan \beta =\frac{\sin (\alpha \pm \beta )}{\cos \alpha \cos \beta }$$
  • $$\cot \alpha \pm \cot \beta =\frac{\sin (\beta \pm \alpha )}{\sin \alpha \sin \beta }$$
  • $$\sin \alpha +\cos \alpha =\sqrt{2}\sin \left ( \alpha +\frac{\pi }{4} \right )=\sqrt{2}\cos \left ( \frac{\pi }{4}-\alpha \right )$$
  • $$\sin \alpha -\cos \alpha =\sqrt{2}\sin \left ( \alpha -\frac{\pi }{4} \right )=-\sqrt{2}\cos \left ( \frac{\pi }{4}-\alpha \right )$$

Trigonometrijske funkcije izražene preko drugih trigonometrijskih funkcija

\(\sin \alpha \) \(\cos \alpha \) \(\tan \alpha \) \(\cot \alpha \)
\(\sin \alpha =\) \(\pm \sqrt{1-\cos ^2\alpha }\) \(\pm \frac{\tan \alpha }{\sqrt{1+\tan ^2\alpha }}\) \(\pm \frac{1}{\sqrt{1+\cot ^2\alpha }}\)
\(\cos \alpha =\) \(\pm \sqrt{1-\sin ^2\alpha }\) \(\pm \frac{1}{\sqrt{1+\tan ^2\alpha }}\) \(\pm \frac{\cot \alpha }{\sqrt{1+\cot ^2\alpha }}\)
\(\tan \alpha =\) \(\pm \frac{\sin \alpha }{\sqrt{1-\sin ^2\alpha }}\) \(\pm \frac{\sqrt{1-\cos ^2\alpha }}{\cos \alpha }\) \(\frac{1}{\cot \alpha }\)
\(\cot \alpha =\) \(\pm \frac{\sqrt{1-\sin ^2\alpha }}{\sin \alpha }\) \(\pm \frac{\cos \alpha }{\sqrt{1-\cos ^2\alpha }}\) \(\frac{1}{\tan \alpha }\)

Univerzalna smena

Za \(\tan \frac{x}{2}=t\) važi:

  • $$\sin x=\frac{2t}{1+t^2}$$
  • $$\cos x=\frac{1-t^2}{1+t^2}$$
  • $$\tan x=\frac{2t}{1-t^2}$$
  • $$\cot x=\frac{1-t^2}{2t}$$

Sinusna i kosinusna teorema

Za proizvoljan trougao čije su stranice dužina \(a,b\) i \(c\) i uglovi \(\alpha ,\beta\) i \(\gamma \), važi:

  • $$\frac{a}{\sin \alpha }=\frac{b}{\sin \beta }=\frac{c}{\sin \gamma }$$
  • $$c^2=a^2+b^2-2ab\cos \gamma $$
  • $$a^2=b^2+c^2-2bc\cos \alpha $$
  • $$b^2=a^2+c^2-2ac\cos \beta $$

Posledice:

  • $$\frac{a-b}{a+b}=\frac{\tan \frac{\alpha +\beta }{2}}{\tan \frac{\alpha -\beta }{2}}$$
  • $$\frac{a+b}{c}=\frac{\cos \frac{\alpha +\beta }{2}}{\sin \frac{\gamma }{2}}$$
  • $$\frac{a-b}{c}=\frac{\sin \frac{\alpha -\beta }{2}}{\cos \frac{\gamma }{2}}$$
  • $$\sin \alpha =\frac{2}{bc}\sqrt{s(s-a)(s-b)(s-c)}, s=\frac{a+b+c}{2}$$

Trigonometrija uglova trougla

Ako je \(\alpha +\beta +\gamma =180^{\circ}\), tada važe formule:

  • $$\sin \alpha +\sin \beta +\sin \gamma =4\cos \frac{\alpha }{2}\cos \frac{\beta }{2}\cos \frac{\gamma }{2}$$
  • $$\cos \alpha +\cos \beta +\cos \gamma =4\sin \frac{\alpha }{2}\sin \frac{\beta }{2}\sin \frac{\gamma }{2}+1$$
  • $$\sin \alpha +\sin \beta -\sin \gamma =4\sin \frac{\alpha }{2}\sin \frac{\beta }{2}\cos \frac{\gamma }{2}$$
  • $$\cos \alpha +\cos \beta -\cos \gamma =4\cos \frac{\alpha }{2}\cos \frac{\beta }{2}\sin \frac{\gamma }{2}-1$$
  • $$\sin^2 \alpha +\sin^2 \beta +\sin^2 \gamma =2\cos \alpha \cos \beta \cos \gamma +2$$
  • $$\sin^2 \alpha +\sin^2 \beta -\sin^2 \gamma =2\sin \alpha \sin \beta \cos \gamma $$
  • $$\sin 2\alpha +\sin 2\beta +\sin 2\gamma =4\sin \alpha \sin \beta \sin \gamma$$
  • $$\sin 2\alpha +\sin 2\beta -\sin 2\gamma =4\cos \alpha \cos \beta \sin \gamma$$
  • $$\tan \alpha +\tan \beta +\tan \gamma =\tan \alpha \tan \beta \tan \gamma$$
  • $$\cot \frac{\alpha }{2}+\cot \frac{\beta }{2}+\cot \frac{\gamma }{2}=\cot \frac{\alpha }{2}\cot \frac{\beta }{2}\cot \frac{\gamma }{2}$$
  • $$\cot \alpha \cot \beta +\cot \alpha \cot \gamma + \cot \beta \cot \gamma =1$$