Osnovne trigonometrijske formule

Funkcije jednog kuta

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

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

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

Međusobno izražavanje funkcija

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

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

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

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

Funkcije zbroja i razlike

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

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

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

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

$\sin 2\alpha =2\sin \alpha \cos \alpha ,\sin 3\alpha =3\sin \alpha -4{\sin }^3\alpha ,$

$\cos 2\alpha ={\cos }^2\alpha -{\sin }^2\alpha ,\cos 3\alpha =4{\cos }^3\alpha -3\cos \alpha ,$

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

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

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

Na osnovu ovih formula možemo odrediti predznak trigonometrijskih funkcija po kvadrantima

Kut	0°- 90°	90°- 180°	180°- 270°	270°- 360°
Kvadrant	I.	II.	III.	IV.
sinus	+	+	-	-
kosinus	+	-	-	+
tangens	+	-	+	-
kotangens	+	-	+	-

Zbroj i razlika trigonometrijskih funkcija

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

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

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

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

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

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

Umnožak funkcija

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

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

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

Funkcije polovine kuta

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

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

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

Potenciranje funkcija

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

${\sin }^3\alpha =\frac{1}{4}(3\sin \alpha -\sin 3\alpha ),{\cos }^3\alpha =\frac{1}{4}(\cos 3\alpha +3\cos \alpha ),$ ${\sin }^4\alpha =\frac{1}{8}(\cos 4\alpha -4\cos 2\alpha +3),{\cos }^4\alpha =\frac{1}{8}(\cos 4\alpha +4\cos 2\alpha +3).$