Osnovne trigonometrijske formule

Funkcije jednog kuta

[math]\displaystyle{ \sin ^2\alpha + \cos ^2 \alpha = 1, \quad \frac{\sin \alpha}{\cos \alpha}=\tan \alpha, \quad \sin \alpha \cdot \csc \alpha = 1 }[/math],

[math]\displaystyle{ \sec ^2 \alpha - \tan ^2 \alpha = 1, \qquad \cos \alpha \cdot \sec \alpha = 1 }[/math],

[math]\displaystyle{ \csc ^2 \alpha - \cot ^2 \alpha = 1, \quad \frac{\cos \alpha}{\sin \alpha} = \cot \alpha, \quad \tan \alpha \cdot \cot \alpha = 1 }[/math]

Međusobno izražavanje funkcija

[math]\displaystyle{ \sin \alpha = \sqrt{1 - \cos ^2 \alpha} = \frac{ \tan \alpha}{ \sqrt{ 1 + \tan ^2 \alpha}}, }[/math]

[math]\displaystyle{ \cos \alpha = \sqrt{1- \sin ^2 \alpha}=\frac{1}{\sqrt{1+ \tan ^2 \alpha}} , }[/math]

[math]\displaystyle{ \tan \alpha = \frac{\sin \alpha}{\sqrt{1- \sin ^2\alpha}}=\frac{1}{\cot \alpha}, }[/math]

[math]\displaystyle{ \cot \alpha = \frac{\sqrt{1- \sin ^2\alpha}}{\sin \alpha}= \frac{1}{\tan \alpha}. }[/math]

Funkcije zbroja i razlike

[math]\displaystyle{ \sin ( \alpha \pm \beta )= \sin \alpha \cos \beta \pm \cos \alpha \sin \beta,\, }[/math]

[math]\displaystyle{ \cos (\alpha \pm \beta )= \cos \alpha \cos \beta \mp \sin \alpha \sin \beta, }[/math]

[math]\displaystyle{ \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}. }[/math]

[math]\displaystyle{ \tan2\alpha=\frac{2\tan\alpha}{1-\tan^2\alpha}, \tan3\alpha=\frac{3\tan\alpha-\tan^3\alpha}{1-3\tan^2\alpha}, }[/math]

[math]\displaystyle{ \sin2\alpha = 2\sin\alpha\cos\alpha, \quad \sin3\alpha=3\sin\alpha-4\sin^3\alpha, }[/math]

[math]\displaystyle{ \cos2\alpha = \cos^2\alpha-\sin^2\alpha, \quad \cos3\alpha=4\cos^3\alpha-3\cos\alpha, }[/math]

[math]\displaystyle{ \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}, }[/math]

[math]\displaystyle{ \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}, }[/math]

[math]\displaystyle{ \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}. }[/math]

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

[math]\displaystyle{ \sin\alpha+\sin\beta=2\sin\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}, }[/math]

[math]\displaystyle{ \sin\alpha-\sin\beta=2\cos\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}, }[/math]

[math]\displaystyle{ \cos\alpha+\cos\beta=2\cos\frac{\alpha+\beta}{2}\cos\frac{\alpha-\beta}{2}, }[/math]

[math]\displaystyle{ \cos\alpha-\cos\beta=-2\sin\frac{\alpha+\beta}{2}\sin\frac{\alpha-\beta}{2}, }[/math]

[math]\displaystyle{ \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}, }[/math]

[math]\displaystyle{ \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}. }[/math]

Umnožak funkcija

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

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

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

Funkcije polovine kuta

[math]\displaystyle{ \sin\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{2}}, \quad \cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}, }[/math]

[math]\displaystyle{ \tan\frac{\alpha}{2}=\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}}=\frac{1-\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1+\cos\alpha}, }[/math]

[math]\displaystyle{ \cot\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}}=\frac{1+\cos\alpha}{\sin\alpha}=\frac{\sin\alpha}{1-\cos\alpha}. }[/math]

Potenciranje funkcija

[math]\displaystyle{ \sin^2\alpha=\frac{1}{2}(1-\cos2\alpha), \quad \cos^2\alpha=\frac{1}{2}(1+\cos2\alpha), }[/math]

[math]\displaystyle{ \sin^3\alpha=\frac{1}{4}(3\sin\alpha-\sin3\alpha), \quad \cos^3\alpha=\frac{1}{4}(\cos3\alpha+3\cos\alpha), }[/math] [math]\displaystyle{ \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). }[/math]