Стандартный список тригонометрических формул

1. \( \sin^2\alpha + \cos^2 \alpha = 1 \)
2. \( \sin(\alpha + \beta) = \sin \alpha \cdot \cos \beta + \sin \beta \cdot \cos \alpha \)
3. \( \sin(\alpha - \beta) = \sin \alpha \cdot \cos \beta - \sin \beta \cdot \cos \alpha \)
4. \( \cos(\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta \)
5. \( \cos(\alpha - \beta) = \cos \alpha \cdot \cos \beta + \sin \alpha \cdot \sin \beta \)
6. \( \sin 2\alpha = 2\sin\alpha\cos\alpha \)
7. \( \cos 2\alpha = \cos^2 \alpha - \sin^2 \alpha \)
8. \( \cos 2\alpha = 2 \cos^2 \alpha - 1 \)
9. \( \cos 2\alpha = 1 - 2 \sin^2 \alpha \)
10. \( \sin^2 \alpha = \dfrac{1 - \cos 2\alpha}{2} \)
11. \( \cos^2 \alpha = \dfrac{1 + \cos 2\alpha}{2} \)
12. \( \cos \alpha \cdot \cos \beta = \dfrac{1}{2} \bigg( \cos(\alpha + \beta) + \cos(\alpha - \beta) \bigg) \)
13. \( \sin \alpha \cdot \sin \beta = \dfrac{1}{2} \bigg( \cos(\alpha - \beta) - \cos(\alpha + \beta) \bigg) \)
14. \( \sin \alpha \cdot \cos \beta = \dfrac{1}{2} \bigg( \sin(\alpha + \beta) + \sin(\alpha - \beta) \bigg) \)
15. \( \cos \alpha \cdot \sin \beta = \dfrac{1}{2} \bigg( \sin(\alpha + \beta) - \sin(\alpha - \beta) \bigg) \)
16. \( \sin \alpha + \sin \beta = 2 \sin \left( \dfrac{\alpha + \beta}{2} \right) \cos \left( \dfrac{\alpha - \beta}{2} \right) \)
17. \( \sin \alpha - \sin \beta = 2 \cos \left( \dfrac{\alpha + \beta}{2} \right) \sin \left( \dfrac{\alpha - \beta}{2} \right) \)
18. \( \cos \alpha + \cos \beta = 2 \cos \left( \dfrac{\alpha + \beta}{2} \right) \cos \left( \dfrac{\alpha - \beta}{2} \right) \)
19. \( \cos \alpha - \cos \beta = -2 \sin \left( \dfrac{\alpha + \beta}{2} \right) \sin \left( \dfrac{\alpha - \beta}{2} \right) \)
20. \( a \sin \alpha + b \cos \alpha = \sqrt{a^2 + b^2} \cdot \sin(\alpha + \varphi) \), где \(\sin\varphi =\dfrac{b}{a^2+b^2}\) и \(\cos\varphi =\dfrac{a}{a^2+b^2}\)