Some Important Formulas

1. Binomial theorem \begin{align} (a+b)^n=a^n+n a^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+\frac{n(n-1)(n-2)}{3!}a^{n-3}b^3+\cdots \end{align}

2. ${\displaystyle {\log }_c(AB)={\log }_{c}A+{\log }_{c}B}$

3. ${\displaystyle {\log }_c\left(A^n\right)=n{\log }_{c}A}$

4. ${\displaystyle {\log }_c\sqrt[n]{A}={\log }_c\left(A^{\frac{1}{n}}\right)=\frac{1}{n}{\log }_{c}A}$

5. ${\displaystyle {\log }_c\frac{A}{B}={\log }_{c}A-{\log }_{c}B}$

6. ${\displaystyle {\log }_{c}A=\frac{{\log }_{b}A}{{\log }_{b}c}}$

7. ${\displaystyle \tan x=\frac{\sin x}{\cos x}}$

8. $\cot x={\displaystyle \frac{1}{\tan x}}={\displaystyle \frac{\cos x}{\sin x}}$.

9. $\sec x={\displaystyle \frac{1}{\cos x}}$

10. $\csc x={\displaystyle \frac{1}{\sin x}}$

11. $\sin (-x)=-\sin x$

12. $\cos (-x)=\cos x$

13. $\tan (-x)=-\tan x$

14. ${\sin }^{2}x+{\cos }^{2}x=1$

15. $\sin (A\pm B)=\sin A\cos B\pm \cos A\sin B$

16. $\cos (A\pm B)=\cos A\cos B\mp \sin A\sin B$

17. $\tan (A+B)={\displaystyle \frac{\tan A+\tan B}{1-\tan A\tan B}}$

18. $\tan (A-B)={\displaystyle \frac{\tan A-\tan B}{1+\tan A\tan B}}$

19. $\sin 2x=2\sin x\cos x$

20. $\cos 2x={\cos }^{2}x-{\sin }^{2}x$    or
    $\cos 2x=1-2{\sin }^{2}x$    or
    $\cos 2x=2{\cos }^{2}x-1$

21. $\tan 2x={\displaystyle \frac{2\tan x}{1-{\tan }^{2}x}}$

22. ${\sin }^{2}x={\displaystyle \frac{1-\cos 2x}{2}}$

23. ${\cos }^{2}x={\displaystyle \frac{1+\cos 2x}{2}}$

24. $\cos ({\displaystyle \frac{\pi }{2}}-x)=\sin x$

25. $\sin ({\displaystyle \frac{\pi }{2}}-x)=\cos x$

26. $\cot ({\displaystyle \frac{\pi }{2}}-x)=\tan x$

27. $\tan ({\displaystyle \frac{\pi }{2}}-x)=\cot x$

28. $\sin (\pi -x)=\sin x$

29. $\cos (\pi -x)=-\cos x$

30. $\tan (\pi -x)=-\tan x$

31. $\sin A+\sin B=2\sin {\displaystyle \frac{A+B}{2}}\cos {\displaystyle \frac{A-B}{2}}$

32. $\cos A+\cos B=2\cos {\displaystyle \frac{A+B}{2}}\cos {\displaystyle \frac{A-B}{2}}$

33. $\sin M\cos N={\displaystyle \frac{1}{2}}[\sin (M-N)+\sin (M+N)]$

34. $\sin M\sin N={\displaystyle \frac{1}{2}}[\cos (M-N)-\cos (M+N)]$

35. $\cos M\cos N={\displaystyle \frac{1}{2}}[\cos (M-N)+\cos (M+N)]$

36. If $\theta$ is the angle between two lines whose slopes are $m_1$ and $m_2$, then $$\tan \theta =\frac{m_1-m_2}{1+m_1{m}_2}$$

37. Two lines whose slopes are $m_1$ and $m_2$ are parallel if $m_1=m_2$ and are perpendicular if $m_1=-{\displaystyle \frac{1}{m_2}}$.

38. Transforming from polar coordinates to rectangular coordinates $$x=r\cos \theta ,\text{and}y=r\sin \theta$$

39. The area of a triangle with length sides $a$, $b$, and $c$ is $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where $s$ is half the perimeter or $s={\displaystyle \frac{a+b+c}{2}}$.

40. Volume of a sphere of radius $r$: $V={\displaystyle \frac{4}{3}}\pi r^3$

41. Surface area of a sphere of radius $r$: $A=4\pi r^2$

42. Volume of a rectangular pyramid $V=lwh/3$

    

43. Volume of a frustum of pyramid of height $h$ and of base areas $A$ and $a$: $$V=\frac{h}{3}(A+a+\sqrt{Aa})$$

    

44. Volume of a cone of radius $r$ and height $h$ $V=\frac{1}{3}h\pi r^2$

45. Lateral area of a cone of radius $r$ and height $h$: $A_L=\pi r\sqrt{h^2+r^2}$. This can also be written as $A_L=\pi rl$, where $l$ is its slant height.