Some Important Formulas

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. \(\displaystyle \log_c\frac{A}{B}=\log_cA-\log_c B\)

  6. \(\displaystyle \log_c A=\dfrac{\log_b A}{\log_b c}\)

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

  8. \(\cot x=\dfrac{1}{\tan x}=\dfrac{\cos x}{\sin x}\).

  9. \(\sec x=\dfrac{1}{\cos x}\)

  10. \(\csc x=\dfrac{1}{\sin x}\)

  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. \(\tan(A+B)=\dfrac{\tan A+\tan B}{1-\tan A\tan B}\)

  18. \(\tan(A-B)=\dfrac{\tan A-\tan B}{1+\tan A\tan B}\)

  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. \(\tan 2x=\dfrac{2\tan x}{1-\tan^2x}\)

  22. \(\sin^2 x=\dfrac{1-\cos 2x}{2}\)

  23. \(\cos^2 x=\dfrac{1+\cos 2x}{2}\)

  24. \(\cos\left(\dfrac{\pi}{2}-x\right)=\sin x\)

  25. \(\sin\left(\dfrac{\pi}{2}-x\right)=\cos x\)

  26. \(\cot\left(\dfrac{\pi}{2}-x\right)=\tan x\)

  27. \(\tan\left(\dfrac{\pi}{2}-x\right)=\cot x\)

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

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

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

  31. \(\sin A+\sin B=2\sin\dfrac{A+B}{2}\cos\dfrac{A-B}{2}\)

  32. \(\cos A+\cos B=2\cos\dfrac{A+B}{2}\cos\dfrac{A-B}{2}\)

  33. \(\sin M \cos N=\dfrac{1}{2}\left[\sin(M-N)+\sin(M+N)\right]\)

  34. \(\sin M \sin N=\dfrac{1}{2}\left[\cos(M-N)-\cos(M+N)\right]\)

  35. \(\cos M \cos N=\dfrac{1}{2}\left[\cos(M-N)+\cos(M+N)\right]\)

  36. If \(\theta\) is the angle between two lines whose slopes are \(m_1\) and \(m_2\), then \[\tan\theta=\frac{m_1-m_2}{1+m_1m_2}\]

  37. Two lines whose slopes are \(m_1\) and \(m_2\) are parallel if \(m_1=m_2\) and are perpendicular if \(m_1=-\dfrac{1}{m_2}\).

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

  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 \(s=\dfrac{a+b+c}{2}\).

  40. Volume of a sphere of radius \(r\): \(V=\dfrac{4}{3}\pi r^3\)

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

  42. Volume of a rectangular pyramid \(V=lwh/3\)

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

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  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 \(A_L=\pi r l\), where \(l\) is its slant height.

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