Formulas for reference

For the convenience of the student we give the following list of elementary formulas from Algebra, Geometry, Trigonometry, and Analytic Geometry.

1. Binomial Theorem ($n$ being a positive integer):
   $$\begin{matrix} (a + b)^n = a^n + na^{n-1}b &+& \frac{n(n - 1)}{2!... ... - 1)(n - 2)\cdots(n - r + 2)}{(r - 1)!}a^{n-r+1}b^{r-1} + \cdots \end{matrix}$$

2. $n! = 1 \cdot 2 \cdot 3 \cdot 4 \cdots (n - 1)n$.

3. In the quadratic equation $ax^2 + bx + c = 0$,

           when $b^2-4ac > 0$, the roots are real and distinct;

           when $b^2-4ac = 0$, the roots are real and equal;

           when $b^2-4ac < 0$, the roots are complex.

4. When a quadratic equation is reduced to the form $x^2 + px + q = 0$,

           $p$ = sum of roots with sign changed, and

           $q$ = product of roots.

5. In an arithmetical series, $a$, $a+d$, $a+2d$, ...,
   $$s = \sum_{i=0}^{n-1} a + id = \frac{n}{2}[2a + (n-1)d].$$

6. In a geometrical series, $a$, $ar$, $ar^2$, ...,
   $$s = \sum_{i=0}^{n-1} ar^{i} = \frac{a(r^n - 1)}{r - 1}.$$

7. $\log ab = \log a + \log b$.

8. $\log \frac{a}{b} = \log a - \log b$.

9. $\log a^n = n\log a$.

10. $\log \sqrt[n]{a} = \frac{1}{n} \log a$.

11. $\log 1 = 0$.

12. $\log e = 1$.

13. $\log \frac{1}{a} = -\log a$.

14. ^1.1Circumference of circle = $2 \pi\, r$.

15. Area of circle = $\pi\, r^2$.

16. Volume of prism = $Ba$.

17. Volume of pyramid = $\frac{1}{3} Ba$.

18. Volume of right circular cylinder = $\pi\, r^2a$.

19. Lateral surface of right circular cylinder = $2 \pi\, ra$.

20. Total surface of right circular cylinder = $2 \pi\, r(r + a)$.

21. Volume of right circular cone = $2 \pi\, r(r + a)$.

22. Lateral surface of right circular cone = $\pi\ rs$.

23. Total surface of right circular cone = $\pi\ r(r + s)$.

24. Volume of sphere = $\frac{4}{3}\pi\ r^3$.

25. Surface of sphere = $4\pi\ r^2$.

26. $\sin x = \frac{1}{\csc x}$;

    $\cos x = \frac{1}{\sec x}$;

    $\tan x = \frac{1}{\cot x}$.

27. $\tan x = \frac{\sin{x}}{\cos{x}}$;

    $\cot{x} = \frac{\cos{x}}{\sin{x}}$.

28. $\sin^2 x + \cos^2 x = 1$;

    $1 + \tan^2 x = \sec^2 x$;

    $1 + \cot^2 x = \csc^2 x$.

29. $\sin x = \cos \left ( \frac{\pi}{2} - x \right )$;

    $\cos x = \sin \left ( \frac{\pi}{2} - x \right)$;

    $\tan x = \cot \left ( \frac{\pi}{2} - x \right )$.

30. $\sin(\pi\ - x) = \sin x$;

    $\cos(\pi\ - x) = -\cos x$;

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

31. $\sin (x + y) = \sin x \cos y + \cos x \sin y$.

32. $\sin (x-y) = \sin x \cos y-\cos x \sin y$.

33. $\cos(x \pm y) = \cos x \cos y +\mp \sin x \sin y$.

34. $\tan(x + y) = \frac{\tan x + \tan y}{1 - \tan x \tan y}$.

35. $\tan(x - y) = \frac{\tan x - \tan y}{1 + \tan x \tan y}$.

36. $\sin 2x = 2 \sin x \cos x$; $\cos 2x = \cos^2 x - \sin^2 x$; $\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}$.

37. $\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}$; $\cos x = \cos^2 \frac{x}{2} - \sin^2 \frac{x}{2}$; $\tan x = \frac{2 \tan \frac{1}{2} x}{1 - \tan^2 \frac{1}{2} x}$.

38. $\cos^2 x = \frac{1}{2} + \frac{1}{2} \cos 2x$; $\sin^2 x = \frac{1}{2} - \frac{1}{2} \cos 2x$.

39. $1 + \cos x = 2 \cos^2 \frac{x}{2}$; $1 - \cos x = 2 \sin^2 \frac{x}{2}$.

40. $\sin \frac{x}{2} = \pm \sqrt{ \frac{1 - \cos x}{2} }$; $\cos x/2 = \pm \sqrt{ \frac{1 + \cos x}{2} }$; $\tan \frac{x}{2} = \pm \sqrt{ \frac{1 - \cos x}{1 + \cos x}}$.

41. $\sin x + \sin y = 2 \sin \frac{1}{2} (x + y) \cos \frac{1}{2} (x - y)$.

42. $\sin x - \sin y = 2 \cos \frac{1}{2} (x + y) \sin \frac{1}{2} (x - y)$.

43. $\cos x + \cos y = -2 \cos \frac{1}{2} (x + y) \cos \frac{1}{2} (x - y)$.

44. $\cos x - \cos y = -2 \sin \frac{1}{2} (x + y) \sin \frac{1}{2} (x - y)$.

45. $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$; Law of Sines.

46. $a^2 = b^2 + c^2 − 2bc\cos A$; Law of Cosines.

47. $d = \sqrt{ (x_1 - x_2)^2 + (y_1 - y_2)^2}$; distance between points $(x_1,y_1)$ and $(x_2,y_2)$.

48. $d = \frac{Ax_1 + By_1 + C}{\pm \sqrt{A^2 + B^2}}$; distance from line $Ax + By + C = 0$ to $(x_1,y_1)$.

49. $x = \frac{x_1 + x_2}{2}$, $y = \frac{y_1 + y_2}{2}$; coordinates of middle point.

50. $x = x_0 + x'$, $y = y_0 + y'$; transforming to new origin $(x_0,y_0)$.

51. $x = x' \cos \theta\ - y' \sin \theta\$, $y = x' \sin \theta\ + y' \cos \theta$; transforming to new axes making the angle theta with old.

52. $x = \rho\ \cos \theta\$, $y = \rho\ \sin \theta$; transforming from rectangular to polar coordinates.

53. $\rho\ = \sqrt{x^2 + y^2}$, $\theta\ = \arctan \frac{y}{x}$; transforming from polar to rectangular coordinates.

54. Different forms of equation of a straight line:

    (a)

    $\frac{y - y_1}{x - x_1} = \frac{y_2 - y_1}{x_2 - x_1}$, two-point form;

    (b)

    $\frac{x}{a} + \frac{y}{b} = 1$, intercept form;

    (c)

    $y-y_1 = m(x-x_1)$, slope-point form;

    (d)

    $y = mx + b$, slope-intercept form;

    (e)

    $x \cos \alpha\ + y \sin \alpha\ = p$, normal form;

    (f)

    $Ax + By + C = 0$, general form.

55. $\tan \theta\ = \frac{m_1 - m_2}{1 + m_1 m_2}$, angle between two lines whose slopes are $m_1$ and $m_2$.

            $m_1 = m_2$ when lines are parallel, and

             $m_1 = -\frac{1}{m_2}$ when lines are perpendicular.

56. $(x - \alpha)^2 + (y - \beta)^2 = r^2$, equation of circle with center $(\alpha,\beta)$ and radius $r$.

Many of these facts are already known to Sage:

[fontsize=\scriptsize,fontfamily=courier,fontshape=tt,frame=single,label=\sage]

sage: a,b = var("a,b")
sage: log(sqrt(a))
log(a)/2
sage: log(a/b).simplify_log()
log(a) - log(b)
sage: sin(a+b).simplify_trig()
cos(a)*sin(b) + sin(a)*cos(b)
sage: cos(a+b).simplify_trig()
cos(a)*cos(b) - sin(a)*sin(b)
sage: (a+b)^5
(b + a)^5
sage: expand((a+b)^5)
b^5 + 5*a*b^4 + 10*a^2*b^3 + 10*a^3*b^2 + 5*a^4*b + a^5

``Under the hood'' Sage used Maxima to do this simplification.