Elementarne funkcije

Trigonometrijske funkcije

Osnovne trigonometrijske formule

1. Funkcije jednog ugla
$$\sin^{2} \alpha+\cos^{2} \alpha=1, \frac{\sin \alpha}{\cos \alpha}=\tan \alpha, \sin \alpha \cdot \csc \alpha=1$$ $$\sec^{2} \alpha-\tan^{2} \alpha=1, \cos \alpha \cdot \sec \alpha=1$$ $$\csc^{2} \alpha-\cot^{2} \alpha=1, \frac{\cos \alpha}{\sin \alpha}=\cot \alpha, \tan \alpha \cdot \cot \alpha=1$$
2. Međusobno izražavanje funkcija
$$\sin \alpha=\sqrt{1 − \cos^{2} \alpha}=\frac{\tan \alpha}{1+\tan^{2} \alpha} \cos \alpha=\sqrt{1−\sin^{2} \alpha}=\frac{1}{1+\tan^{2} \alpha} \tan \alpha=\frac{\sin \alpha}{\sqrt{1−\sin^{2} \alpha}}=\frac{1}{\cot \alpha} \cot \alpha=$$ $$=\frac{\sqrt{1−\sin^{2} \alpha}}{\sin \alpha}=\frac{1}{\tan \alpha}$$
3. Funkcije zbira i razlike
$$\sin(\alpha\pm\beta)=\sin \alpha \cos \beta\pm\cos \alpha \sin \beta$$ $$\cos(\alpha\pm\beta)=\cos \alpha \cos \beta∓\sin \alpha \sin \beta$$ $$\tan(\alpha\pm\beta)=\frac{\tan \alpha\pm\tan \beta}{1∓\tan \alpha \tan \beta}$$ $$\cot(\alpha\pm\beta)=\frac{\cot \alpha \cot \beta∓1}{\cot \beta\pm\cot \alpha}$$
4. Funkcije dvostrukog ugla
$$\sin 2\alpha=2 \sin \alpha \cos \alpha, \sin 3\alpha=3 \sin \alpha−\sin^{3} \alpha$$ $$\cos 2\alpha=\cos^{2} \alpha−\sin^{2} \alpha, \cos 3\alpha=4 \cos^{3} \alpha−3 \cos \alpha$$ $$\tan 2\alpha=\frac{2 \tan \alpha}{1−\tan^{2} \alpha}, \tan 3\alpha=\frac{3 \tan \alpha−\tan^{3} \alpha}{1−3 \tan^{2} \alpha}$$ $$\cot 2\alpha=\frac{\cot^{2} \alpha−1}{2 \cot \alpha}, \cot 4\alpha=\frac{\cot^{4} \alpha−6 \cot^{2} \alpha+1}{4 \cot^{3} \alpha−4 \cot \alpha}$$ $$\tan 4\alpha=\frac{4 \tan \alpha−4 \tan^{3} \alpha}{1−6 \tan^{2} \alpha+\tan^{4} \alpha}, \cot 4\alpha=\frac{\cot^{4} \alpha−6 \cot^{2} \alpha+1}{4 \cot^{3} \alpha−4 \cot \alpha}$$
5. Zbir i razlika funkcija
$$\sin \alpha+\sin \beta=2\sin \frac{\alpha+\beta}{2}\cos \frac{\alpha-\beta}{2}$$ $$\sin \alpha-\sin \beta=2\cos \frac{\alpha+\beta}{2}\sin \frac{\alpha-\beta}{2}$$ $$\cos \alpha+\cos \beta=2\cos \frac{\alpha+\beta}{2}\cos \frac{\alpha-\beta}{2}$$ $$\cos \alpha-\cos \beta=-2\cos \frac{\alpha+\beta}{2}\sin \frac{\alpha-\beta}{2}$$ $$\tan \alpha\pm\tan \beta=\frac{\sin \alpha\pm\beta}{\cos \alpha \cos \beta}, \cot \alpha\pm\cot \beta=\frac{\sin \alpha\pm\beta}{\sin \alpha \sin \beta}$$ $$\tan \alpha+\cot \beta=\frac{\cos \alpha- \beta}{\cos \alpha \sin \beta}, \cot \alpha-\tan \beta=\frac{\cos \alpha+ \beta}{\cos \beta \sin \alpha}$$
6. Proizvod funkcija
$$\sin \alpha \cos \beta=\frac{1}{2}[\sin(\alpha+\beta)+\sin(\alpha−\beta)]$$ $$\cos \alpha \sin \beta=\frac{1}{2}[\sin(\alpha+\beta)-\sin(\alpha−\beta)]$$ $$\cos \alpha \cos \beta=\frac{1}{2}[\cos(\alpha+\beta)+\cos(\alpha−\beta)]$$ $$\sin \alpha \sin \beta=-\frac{1}{2}[\cos(\alpha+\beta)-\cos(\alpha-\beta)]$$
7. Funkcije polovine ugla
$$\sin \frac{\alpha}{2}=\pm\sqrt{\frac{1−\cos \alpha}{2}}$$ $$\cos \frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos \alpha}{2}}$$ $$\tan \frac{\alpha}{2}=\pm\sqrt{\frac{1−\cos \alpha}{1+\cos \alpha}}$$ $$\cot \frac{\alpha}{2}=\pm\sqrt{\frac{1+\cos \alpha}{1−\cos \alpha}}$$
8. Stepenovanje funkcija
$$\sin^{2} \alpha=\frac{1}{2}(1−\cos 2\alpha), \cos^{2} \alpha=\frac{1}{2}(1+\cos 2\alpha)$$ $$\sin^{3} \alpha=\frac{1}{4}(3 \sin \alpha−\sin 3\alpha), \cos^{3}\alpha=\frac{1}{4}(3 \cos \alpha+\cos 3\alpha)$$ $$\sin^{4} \alpha=\frac{1}{8}(\cos 4\alpha−4 \cos 2\alpha+3), \cos^{4} \alpha=\frac{1}{8}(\cos 4\alpha+4 \cos 2\alpha+3)$$