Osnovne trigonometrijske fornule

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1 Funkcije jednog ugla
2 Međusobno izražavanje funkcija
3 Funkcije zbira i razlike
4 Zbir i razlika trigonometrijskih функција
5 Proizvod funkcija
6 Funkcije polovine ugla
7 Stepenovanje funkcija

[uredi] Funkcije jednog ugla

$\sin ^2\alpha + \cos ^2 \alpha = 1, \quad \frac{\sin \alpha}{\cos \alpha}=\tan \alpha, \quad \sin \alpha \cdot \csc \alpha = 1$ ,

$\sec ^2 \alpha - \tan ^2 \alpha = 1, \qquad \cos \alpha \cdot \sec \alpha = 1$ ,

$\csc ^2 \alpha - \cot ^2 \alpha = 1, \quad \frac{\cos \alpha}{\sin \alpha} = \cot \alpha, \quad \tan \alpha \cdot \cot \alpha = 1$

[uredi] Međusobno izražavanje funkcija

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

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

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

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

[uredi] Funkcije zbira i razlike

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

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

$\tan (\alpha \pm \beta )=\frac{\tan \alpha \pm \tan \beta}{1 \mp \tan \alpha \tan \beta}, \quad \cot ( \alpha \pm \beta ) = \frac{\cot \alpha \cot \beta \mp 1}{\cot \beta \pm \cot \alpha}.$

$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}, \tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha},$

$\sin2\alpha = 2\sin\alpha\cos\alpha, \quad \sin3\alpha=3\sin\alpha-4\sin^3\alpha,$

$\cos2\alpha = \cos^2\alpha-\sin^2\alpha, \quad \cos3\alpha=4\cos^3\alpha-3\cos\alpha,$

$\tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}, \quad \tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha},$

$\cot2\alpha=\frac{\cot^2\alpha-1}{2\cot\alpha}, \quad \cot3\alpha=\frac{\cot^3\alpha-3\cot\alpha}{3\cot^2\alpha-1},$

$\tan4\alpha=\frac{4\tan\alpha-4\tan^3\alpha}{1-6\tan^2\alpha+\tan^4\alpha}, \quad \cot4\alpha=\frac{\cot^4\alpha-6\cot^2\alpha+1}{4\cot^3\alpha-4\cot\alpha}.$

Na osnovu ovih formula možemo odrediti predznak trigonometrijskih funkcija po kvadrantima

- align="center"	Kvadrant	0°-90°	90°-180°	180°-270°)
sinus	+	+	-	-
kosinus	+	-	-	+
tangens	+	-	+	-

[uredi] Zbir i razlika trigonometrijskih функција

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

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

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

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

$\tan\alpha\pm\tan\beta=\frac{\sin (\alpha\pm\beta )}{\cos\alpha\cos\beta}, \quad \cot\alpha\pm\cot\beta=\pm\frac{\sin (\alpha\pm\beta)}{\sin\alpha\sin\beta},$

$\tan\alpha+\cot\beta=\frac{\cos (\alpha-\beta)}{\cos\alpha\sin\beta}, \quad \cot\alpha-\tan\beta=\frac{cos (\alpha+\beta)}{\sin\alpha\cos\beta}.$

[uredi] Proizvod funkcija

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

$\cos\alpha\cos\beta=\frac{1}{2}[\cos(\alpha-\beta)+cos(\alpha+\beta)],$

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

[uredi] Funkcije polovine ugla

$\sin\frac{\alpha}{2}=\frac{\sqrt{1-\cos\alpha}}{2}, \quad \cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}},$

$\tan\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha},$

$\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}=\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}.$

[uredi] Stepenovanje funkcija

$\sin^2\alpha=\frac{1}{2}(1-\cos2\alpha), \quad \cos^2\alpha=\frac{1}{2}(1+\cos2\alpha),$

$\sin^3\alpha=\frac{1}{4}(3\sin\alpha-\sin3\alpha), \quad \cos^3\alpha=\frac{1}{4}(\cos3\alpha+3\cos\alpha),$ $\sin^4\alpha=\frac{1}{8}(\cos4\alpha-4\cos2\alpha+3), \quad \cos^4\alpha=\frac{1}{8}(\cos4\alpha+4\cos2\alpha+3).$