Formulas Flashcards
Distance between a point with coordinates (x1, y1) and a line Ax + By + C = 0 is
|Ax1 + By1 + C| / √(A^2 + B^2)
If θ is the angle between two lines, tanθ =
(m1 - m2) / (1 + m1m2), where m1 and m2 are the slopes of the two lines
sum of roots of a quadratic equation =
-b/a
product of roots of a quadratic equation =
c/a
Area of a triangle ABC is
1/2bc * sinA
(Pythagorean identities) sin²x + cos²x =
1
(Pythagorean identities) sec²x =
tan²x + 1
(Pythagorean identities) csc²x =
cot²x + 1
(Double-angle formulas) sin2A =
2sinAcosA
(double-angle formulas) cos2A= (both cos and sin)
cos²A - sin²A
(double-angle formulas) cos2A= (just cos)
2cos²A - 1
(double-angle formulas) cos2A= (just sin)
1 - 2sin²A
(double-angle formulas) tan2A =
2tanA/ (1-tan²A)
if x≥0, then |x| =
x
if x
-x
[x] =
I, where I is an integer and I≤ x
(polar coordinates) x =
rcosθ
(polar coordinates) y =
rsinθ
(polar coordinates) r² =
x² + y²
Distance between a point with coordinates (x1, y1, z1) and a plane with equation Ax + By + Cz + D= 0 is
(Ax1 + By1 + Cz1 + D)/ √(A² + B² + C²)
volume of a cylinder
πr²h
lateral surface area of a cylinder =
2πrh
total surface area of a cylinder =
2πrh + 2πr²
volume of a cone
1/3 * πr²h
lateral surface area of a cone
πr√(r² + h²)
total surface area of a cone
πr√(r² + h²) + πr²
volume of a sphere
4/3 * πr³
surface area of a sphere
4πr²
nCr = (n/r)=
note: n/r is not actually a fraction
n!/ (n-r)!r! = nPr / r!
(arithmetic sequence) nth term =
tn = t1 + (n - 1)d
(arithmetic sequence) sum of n terms =
Sn= n/2 * (t1 + tn)
(geometric sequence) nth term =
tn= t1 * r^(n-1)
(geometric sequence) sum of n terms =
Sn = t1(1 - r^n) / (1 - r)
(geometric sequence) if |r|
lim as n→∞ of Sn = t1/ 1 - r
cofunctions of complementary angles are
equal
cofunction pairs:
sin & cos, sec & csc, tan & cot
the constant in a polynomial equation (e.g. x^4 + 15x^3 + x^2 + 2x + 1) is the
y-intercept
multiplication of matrices is possible only when the number of columns in the matrix on the left equals the number of ….. in that case, the product is a matrix with the number of rows equal to the number of rows in the …. and the number of columns equal to the number of columns in the ….
rows in the matrix on the right/ left-hand matrix; right-hand matrix
to find the inverse of a graph,
fold the graph about the line y = x
even function proof:
f(-x) = f(x)
odd function proof:
f(-x) = -f(x)
when equation is given in the form: Ax + By = C, the slope is… and the y-int is …. If B= 0 and A doesn’t equal 0, then the equation is …
-A/B; C/B; x = C/A, a vertical line
volume of a prism:
V = Bh
surface area of a prism:
SA = Ph + 2B (p is the base perimeter)
surface area of a rectangular solid:
SA = 2lw + 2lh + 2wh
surface area of a cube:
SA = 6s^2
volume of a pyramid:
V= 1/3 Bh
surface area of a pyramid:
SA = 1/2 PL + B (P is the base perimeter, L is the slant height)
diagonal of a rectangular solid:
D= √(l² + w² + h²)
length of diagonal of a cube:
e√3, where e is side length
|x| = b→
x = b or x = -b
|x|
-b
|x| > b→
x b
first few primes:
2, 3, 5, 7, 11, 13, 17, 19, 23
if ad > bc, then
a/b > c/d (this is true when the sign is changed to
is/ of =
% / 100
(x - y)(x + y) =
x² - y²
(x - y)² =
x² - 2xy + y²
(x + y)² =
x² + 2xy + y²
(for triangles) a² + b²
obtuse
(for triangles) a² + b² > c² if and only if angle C is
acute
area of equilateral triangle with side s,
A = (s²√3) / 4
if 2 triangles are similar and if k is the ratio of similitude, the ratio of their areas is
k²
in any polygon, the sum of the measures of the exterior angles, taking one at each vertex, is
360 degrees
(parallelograms) opposite sides are
parallel and congruent
(parallelograms) opposite angles are
congruent
(parallelograms) the sum of the measures of any pair of consecutive angles is
180 degrees
(parallelograms) a diagonal divides the parallelogram into
two congruent triangles
(parallelograms) the two diagonals
bisect each other
(rectangles) has all the properties of
parallelograms
(rectangles) the diagonals of a rectangle have the same
length
(rhombus) has all the properties of
parallelograms
(rhombus) the length of each side is
the same
(rhombus) the two diagonals are
perpendicular
(rhombus) the diagonals are angle
bisectors
squares have all the properties of
parallelograms, rectangles, and rhombi
area of a trapezoid:
A = 1/2 (b1 + b2)h
the measure of an inscribed angle of a circle is
1/2 the measure of its intercepted arc
the measure of the angle formed by 2 tangents drawn from the same point is
1/2 the difference of the two intercepted arcs
percentiles divide a set of data into
100 roughly equal groups
there are … 4-digit integers
9000
