Calculus AB and Calculus BC
Appendix: Formulas and Theorems for Reference
1. QUADRATIC FORMULA. The roots of the quadratic equation
ax2 + bx + c = 0 (a ≠ 0)
are given by
2. BINOMIAL THEOREM. If n is a positive integer, then
3. REMAINDER THEOREM. If the polynomial Q(x) is divided by (x − a) until a constant remainder R is obtained, then R = Q(a). In particular, if a is a root of Q(x) = 0, then Q(a) = 0.
The sum of the angles of a triangle is equal to a straight angle (180°).
In a right triangle,
c2 = a2 + b2.
In the following formulas,
is area of base
height or altitude
central angle (in radians)
6. Parallelogram: A = bh.
7. Circle: C = 2πr, A = πr2.
8. Circular sector:
9. Circular arc: l = rθ.
13. sin2 θ + cos2 θ = 1.
14. 1 + tan2 θ = sec2 θ.
15. 1 + cot2 θ = csc2 θ.
SUM AND DIFFERENCE FORMULAS
16. sin (α ± β) = sin α cos β ± cos α sin β.
17. cos (α ± β) = cos α cos β sin α sin β.
19. sin 2α = 2 sin α cos α.
20. cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α.
24. sin (−α) = −sin α: cos (−α) = cos α.
27. sin (π − α) = sin α; cos (π − α) = −cos α.
28. sin (π + α) = −sin α; cos (π + α) = −cos α.
29. LAW OF COSINES. c2 = a2 + b2 − 2ab cos C.
30. LAW OF SINES.
31. The area sin C.
GRAPHS OF TRIGONOMETRIC FUNCTIONS
The four functions sketched above, sin, cos, csc, and sec, all have period 2π.
32. The distance d between two points, P1 (x1, y1) and P2 (x2, y2), is given by
EQUATIONS OF THE STRAIGHT LINE
33. POINT-SLOPE FORM. Through P1 (x1, y1) and with slope m:
y − y1 = m(x − x1).
34. SLOPE-INTERCEPT FORM. With slope m and y-intercept b:
y = mx + b.
35. TWO-POINT FORM. Through P1 (x1, y1) and P2 (x2, y2):
36. INTERCEPT FORM. With x- and y-intercepts of a and b, respectively:
37. GENERAL FORM. Ax + By + C = 0, where A and B are not both zero. If B ≠ 0, the slope is the y-intercept, the x-intercept,
DISTANCE FROM POINT TO LINE
38. Distance d between a point P(x1, y1) and the line Ax + By + C = 0 is
EQUATIONS OF THE CONICS
39. With center at (0, 0) and radius r: x2 + y2 = r2.
40. With center at (h, k) and radius r: (x − h)2 + (y − k)2 = r2.
41. With vertex at (0, 0) and focus at (p, 0): y2 = 4px.
42. With vertex at (0, 0) and focus at (0, p): x2 = 4py.
With vertex at (h, k) and axis
43. parallel to x-axis, focus at (h + p, k): (y − k)2 = 4p(x − h).
44. parallel to y-axis, focus at (h, k + p): (x − h)2 = 4p(y − k).
With major axis of length 2a, minor axis of length 2b, and distance between foci of 2c:
45. Center at (0, 0), foci at (±c, 0), and vertices at (±a, 0):
46. Center at (0, 0), foci at (0, ±c), and vertices at (0, ±a):
47. Center at (h, k), major axis horizontal, and vertices at (h ± a, k):
48. Center at (h, k), major axis vertical, and vertices at (h, k ± a):
For the ellipse, a2 = b2 + c2, and the eccentricity which is less than 1.
With real (transverse) axis of length 2a, imaginary (conjugate) axis of length 2b, and distance between foci of 2c:
49. Center at (0, 0), foci at (±c, 0), and vertices at (±a, 0):
50. Center at (0, 0), foci at (0, ±c), and vertices at (0, ±a):
51. Center at (h, k), real axis horizontal, vertices at (h ± a, k):
52. Center at (h, k), real axis vertical, vertices at (h, k± a):
For the hyperbola, c2 = a2 + b2, and eccentricity which is greater than 1.
RELATIONS WITH RECTANGULAR COORDINATES
53. x = r cos θ;
y = r sin θ;
r2 = x2 + y2;
SOME POLAR EQUATIONS
54. r = a circle, center at pole, radius a.
55. r = 2a cos θ circle, center at (a, 0), radius a.
56. r = 2a sin θ circle, center at (0, a), radius a.
roses (four leaves)
59. r = cos 2θ.
60. r = sin 2θ.
cardioids (specific examples below)
61. r = a (1 ± cos θ).
62. r = a (1 ± sin θ).
63. r2 = cos 2θ, lemniscate, symmetric to the x-axis.
64. r = θ, (double) spiral of Archimedes
65. rθ = a (θ > 0), hyperbolic (or reciprocal) spiral
EXPONENTIAL AND LOGARITHMIC FUNCTIONS
f (x) = ex and f −1(x) = ln x are inverses of each other:
f −1(f (x)) = f (f −1(x)) = x;
ln ex = eln x = x(x > 0).