Формулаҳои асосии тригонометрӣ
| № | Формула | Қиматҳои имконпазири агрумент |
| 1 | \(\operatorname{sin}^2 \alpha + \operatorname{cos}^2 \alpha = 1\) | \(\forall \alpha\) |
| 2 | \(\operatorname{tg}^2 \alpha + 1 = \frac{1}{\cos^2 \alpha} = \operatorname{sec}^2 \alpha\) | \(\alpha \neq \frac{\pi}{2} + \pi n, n \in \mathbb Z\) |
| 3 | \(\operatorname{ctg}^2 \alpha + 1 = \frac{1}{\sin^2 \alpha} = \operatorname{cosec}^2 \alpha\) | \(\alpha \neq \pi n, n \in \mathbb Z\) |
| 4 | \(\operatorname{tg} \alpha \cdot \operatorname{ctg} \alpha = 1\) | \(\alpha \neq \frac{\pi n}{2}, n \in \mathbb Z\) |
Формулаҳои ҷамъ ва тарҳи аргументҳо
| № | Формула |
| 1 | \(\sin \left( \alpha \pm \beta \right) = \sin \alpha \cos \beta \pm \cos \alpha \sin \beta \) |
| 2 | \(\cos \left( \alpha \pm \beta \right) = \cos \alpha \cos \beta \mp \sin \alpha \sin \beta \) |
| 3 | \(\operatorname{tg} \left( \alpha \pm \beta \right) = \frac{ \operatorname{tg} \alpha \pm \operatorname{tg} \beta}{1 \mp \operatorname{tg} \alpha \operatorname{tg}\beta}\) |
| 4 | \(\operatorname{ctg} \left( \alpha \pm \beta \right) = \frac{ \operatorname{ctg} \alpha \operatorname{ctg} \beta \mp 1}{\operatorname{ctg} \beta \pm \operatorname{ctg}\alpha}\) |
Формулаҳои кунҷҳои дучанда
| № | Формула |
| 1 | \(\operatorname{sin} 2 \alpha = 2 {\sin \alpha}{\cos \alpha}\) |
| 2 | \(\operatorname{cos} 2 \alpha = {\cos^2 \alpha} - {\sin^2 \alpha}\) \(\operatorname{cos} 2 \alpha = 2 {\cos^2 \alpha} - 1 = 1 - 2 {\sin^2 \alpha}\) |
| 3 | \(\operatorname{tg} 2 \alpha = \frac{2 \operatorname{tg} \alpha}{1 - \operatorname{tg}^2 \alpha}\) |
| 4 | \(\operatorname{ctg} 2 \alpha = \frac{\operatorname{ctg}^2 \alpha - 1}{2 \operatorname{ctg} \alpha}\) |
Формулаҳои кунҷҳои сечанда
| № | Формула |
| 1 | \(\sin 3\alpha = 3 \sin \alpha - 4 \sin^3\alpha \,\) |
| 2 | \(\cos 3\alpha = 4 \cos^3\alpha - 3 \cos \alpha \,\) |
| 3 | \(\operatorname{tg} 3\alpha = \frac{3 \operatorname{tg}\alpha - \operatorname{tg}^3\alpha}{1 - 3 \operatorname{tg}^2\alpha}\) |
| 4 | \(\operatorname{ctg} 3\alpha = \frac{3 \operatorname{ctg}\alpha - \operatorname{ctg}^3\alpha}{1 - 3 \operatorname{ctg}^2\alpha}\) |
Формулаҳои паст кардани дараҷа
| № | Синус |
| 1 | \(\sin^2\alpha = \frac{1 - \cos 2\alpha}{2}\) |
| 2 | \(\sin^3\alpha = \frac{3 \sin\alpha - \sin 3\alpha}{4}\) |
| 3 | \(\sin^4\alpha = \frac{3 - 4 \cos 2\alpha + \cos 4\alpha}{8}\) |
| 4 | \(\sin^5\alpha = \frac{10 \sin\alpha - 5 \sin 3\alpha + \sin 5\alpha}{16}\) |
| № | Косинус |
| 5 | \(\cos^2\alpha = \frac{1 + \cos 2\alpha}{2}\) |
| 6 | \(\cos^3\alpha = \frac{3 \cos\alpha + \cos 3\alpha}{4}\) |
| 7 | \(\cos^4\alpha = \frac{3 + 4 \cos 2\alpha + \cos 4\alpha}{8}\) |
| 8 | \(\cos^5\alpha = \frac{10 \cos\alpha + 5 \cos 3\alpha + \cos 5\alpha}{16}\) |
| № | Ҳосили зарб |
| 9 | \(\sin^2\alpha \cos^2\alpha = \frac{1 - \cos 4\alpha}{8}\) |
| 10 | \(\sin^3\alpha \cos^3\alpha = \frac{3\sin 2\alpha - \sin 6\alpha}{32}\) |
| 11 | \(\sin^4\alpha \cos^4\alpha = \frac{3-4\cos 4\alpha + \cos 8\alpha}{128}\) |
| 12 | \(\sin^5\alpha \cos^5\alpha = \frac{10\sin 2\alpha - 5\sin 6\alpha + \sin 10\alpha}{512}\) |
Формулаҳои табдилдиҳии зарби функсияҳо ба сумма
| № | Формула |
| 1 | \(\sin \alpha \sin \beta = \frac{ \cos ( \alpha - \beta) - \cos ( \alpha + \beta)}{2}\) |
| 2 | \(\sin \alpha \cos \beta = \frac{ \sin ( \alpha - \beta) + \sin ( \alpha + \beta) }{2}\) |
| 3 | \(\cos \alpha \cos \beta = \frac{ \cos ( \alpha - \beta) + \cos ( \alpha + \beta)}{2}\) |
Формулаҳои табдилдиҳии суммаи функсияҳо ба зарб
| № | Формула |
| 1 | \(\sin \alpha \pm \sin \beta = 2 \sin \frac{ \alpha \pm \beta}{2} \cos \frac{ \alpha \mp \beta}{2}\) |
| 2 | \(\cos \alpha + \cos \beta = 2 \cos \frac{ \alpha + \beta}{2} \cos \frac{ \alpha - \beta}{2}\) |
| 3 | \(\cos \alpha - \cos \beta = - 2 \sin \frac{ \alpha + \beta}{2} \sin \frac{ \alpha - \beta}{2}\) |
| 4 | \(\operatorname{tg} \alpha \pm \operatorname{tg} \beta = \frac{ \sin ( \alpha \pm \beta)}{ \cos \alpha \cos \beta}\) |
| 5 | \(\operatorname{ctg} \alpha \pm \operatorname{ctg} \beta = \frac{ \sin ( \beta \pm \alpha)}{ \sin \alpha \sin \beta}\) |
