Pre-Calculus Formulas, Pre-Calculus formulas
1 + (cot x)²
(csc x)²
vertex
(h,k); h=(-b/2a), plug h into original equation to find k
1 + (tan x)²
(sec x)²
ellipse
(x²/a²) + (y²/b²) = 1
hyperbola
(x²/a²) - (y²/b²) = 1
hyperbola asymptotes formula
(y-y₁)=m(x-x₁)
ln 1
0
(cos x)² + (sin x)²
1
ln e
1
Increasing Annuity
A = P(1 + r/n) ^ (nt)
odd function
A function that is symmetric with respect to the origin
Geometric Sequences and Series
A sequence is geometric if the ratios of consecutive terms are the same. So, the sequence a1, a2, a3, a4, ... , an, .... is geometric if there is a number r such that a2/a1 = a3/a2 = a4/a3 = ... r, r not equal to zero The number r is the common ratio of the sequence. a2 = a1 * r
Compounded Annuity
A=Pe^rt
Annuity
A=p(1+(r/n))^nt; where p = principle, r = rate, n = number of times interest is compounded each time, and t is time
Series
Consider the infinite sequence a1, a2, a3, ... ai, ... The sum of the first n terms of the sequence is called a finite series or the partial sum of the sequence and is denoted by n a1 + a2 + a3 + ... + an = Σ (ai) i = 1
Polar Coordinates
Each point in the polar plane is assigned the point (r, θ). Related to the rectangular coordinates, x = r cos θ y = r sin θ and tan θ = y/x r^2 = x^2 + y^2
Nth Root of a Complex Number
For a positive integer n, the complex number z = r(cosθ + i*sinθ) has exactly n distinct nth roots given by nth root of r[cos(θ + 2πk)/n + i*sin(θ + 2πk)/n] where k = 0, 1, 2, ..., n - 1
Heron's Area Formula
Given any triangle with side lengths a, b, and c, the area of the triangle is given by Area = √s(s-a)(s-b)(s-c) where s = (a + b + c)/(2)
Degree
Highest exponent in the equation
Interest
I=Prt
The Sum of an Infinite Geometric Sequence
If absolute value of r is less than 1, then the infinite geometric series a1 + a1r + a1r^2, + a1r^3 + ... + has the sum ∞ S = Σ (a1r^i = (a1)/(1-r) i = 0 Note that if the absolute value of r is greater than or equal to one, the series does not have a sum.
Factorial Notation
If n is a positive integer, n factorial is defined as n! = 1 * 2 * 3* 4 *** (n - 1) * n As a special case, zero factorial is defined as 0! = 1 2n! = 2(n!) = 2(1 * 2 * 3 *** n) whereas (2n)! = 1 * 2 * 3 *** 2n
Powers of Complex Numbers - DeMoivre's Theorem
If z = r(cosθ + i*sinθ) is a complex number and n is a positive integer, then z^n = [r(cosθ + i*sinθ)]^n z^n = r^n(cosnθ + i*sinnθ)
Multiplicity
Number of times associated factor appears in polynomial.
Law of Sines
Sin A/a = Sin B/b = Sin C/c or it can be written in its reciprocal form, a/Sin A = b/Sin B = c/Sin C
Law of Cosines
Standard Form a^2 = b^2 + c^2 - 2bccosA b^2 = a^2 + c^2 - 2accosB c^2 = a^2 + b^2 - 2abcosC Alternative Form CosA = (b^2 + c^2 - a^2)/(2bc) CosB = (a^2 + c^2 - b^2)/(2ac) CosC = (a^2 + b^2 - c^2)/(2ab)
The Fibonacci Sequence: A Recursive Sequence
The Fibonacci Sequence is defined recursively as follows: a to the zero = 1, a1 = 1, a to the k = a to the k - 2 + a to the k -1, where k is greater than or equal to 2
The nth Term of a Geometric Sequence
The nth term of a geometric sequence has the form an = a1 * r^(n-1) where r is the common ratio of consecutive terms of the sequence. So, every geometric sequence can be written in the following form. a1, a2, a3, a4, a5, ... an, ... a1, a1r, a1r^2, a1r^3, a1r^4, a1r^(n-1)
The nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence has the form an = dn + c where d is the common difference between consecutive terms of the sequence and c = a1 - d
The Sum of a Finite Arithmetic Sequence
The sum of a finite arithmetic sequence with n terms is given by Sn = n/2 (a1 + an)
Infinite Series
The sum of all the terms of the infinite sequence is called an infinite series and is denoted by ∞ a1 + a2 + a3 + ... + ai + ... = Σ (ai) i = 1
The Sum of a Finite Geometric Sequence
The sum of the finite geometric sequence with common ratio r not equal to 1 is given by n Sn = Σ (a1r^(i - 1)) = a1 (1 - r^n/ 1 - r) i = 1
Summation Notation
The sum of the first n terms of a sequence is represented by n Σ (ai) = a1 + a2 + a3 + a4 + a5 + an i=1 where i is called the index of summation, n is the upper limit of summation, and 1 is the lower limit of summation.
Area of an Oblique Triangle
To see when A is obtuse, substitute the reference angle 180 - A for A. Now the height of the triangle is given by h = bsinA Consequently, the area of each triangle is given by Area = 1/2(base)(height) Area = 1/2(c)(bsinA) Area = 1/2(b)(c)sinA, 1/2(a)(b)sinC, 1/2(a)(c)sinB
Trigonometric Form of a Complex Number
a = rcosθ and b = rsinθ where r = √a^2 + b^2 Consequently, you have a + bi = r(cosθ + i*sinθ) tan θ = b/a
a zero of a function
a value of x where the graph of a function intersects the x-axis
Exponential Form to Log
a^y=x => loga x=y
pythagorean theorem
a² + b² = c²
(cos x)/( sin x)
cot x
1 / (tan x)
cot x
1 / (sin x)
csc x
General Form
f(x) = ax²+bx+c
Vertex Form
f(x)=a(x-h)²+k
to find the equation of the inverse of a function...
interchange x and y, then solve the equation for y
log(AB)
log A + log B
log(A/B)
log A - log B
Log to Exponential Form
loga x=y => a^y=x
Change of Base formula
logb X=ln X/ln b, or log X/log b
slope
m = (y₂-y₁)/(x₂-x₁)
log Aⁿ
nlog A
if point (a,b) lies on the inverse function f^(-1), then...
point (b,a) lies on the function f
to graph the inverse of a function...
reflect the graph of the function across the line y = x
1 / (cos x)
sec x
Even/Odd Identities
sin (-x) = - sin x csc (-x) = - csc x cos (-x) = cos x sec (-x) = sec x tan (-x) = - tan x cot (-x) = - cot x
Sum and Difference Formulas
sin (x + y) = sin x cos y + cos x sin y sin (x- y) = sin x cos y - cos x sin y cos (x + y) = cos x cos y - sin x sin y cos (x - y) = cos x cos y + sin x sin y tan (x + y ) = (tan x + tan y)/(1 - tan x tan y) tan (x - y) = (tan x - tan y)/(1 + tan x tan y)
Cofunction Identities
sin (π/2 - x) = cos x cos (π/2 - x) = sin x tan (π/2 - x) = cot x cot (π/2 - x) = tan x sec (π/2 - x) = csc x csc (π/2 - x) = sec x
Multiple Angle and Formulas
sin 2x = 2 sinx cos x cos 2x = cos^2 x - sin^2 x cos 2x = 1 - 2sin^2 x cos 2x = 2cos^2 x -1 tan 2x = (2 tanx)/(1 - tan x) *sin 4x = 2 sin 2x cos 2x *sin 6x = 2 sin 3x cos 3x
Sum-to-Product Formulas
sin x + sin y = 2sin(x + y)/(2) cos(x -y)/(2) sin x - sin y = 2cos(x + y)/(2) sin (x - y)/(2) cos x + cos y = 2cos(x + y)/(2) cos(x - y)/(2) cos x - cos y = -2 sin(x + y)/(2) sin(x - y)/(2)
Reciprocal Identities
sin x = 1/csc x csc x = 1/sin x cos x = 1/sec x sec x = 1/cos x tan x = 1/cot x cot x = 1/tan x
Half-Angle Formulas
sin x/2 = + or - √(1 - cos x)/2 cos x/2 = + or - √1 + cos x)/2 tan x/2 = (1 - cos x)/(sin x) or (sin x)/(1 + cos x)
Triple Angle Formulas
sin3x = 3sinx - 4sin^3 x cos 3x = 4cos^3 x - 3cosx
Power-Reducing Formulas
sin^2 x = (1 - cos 2x)/(2) cos^2 x = (1 + cos 2x) tan^2 x = (1 - cos 2x)/(1 + cos 2x)
Pythagorean Identities
sin^2 x+cos^2 x=1 tan^2 x+1= sec^2 x 1+cot^2 x=csc^2 x
Product-to-Sum Formulas
sinx sin y = 1/2 [cos(x - y) - cos(x + y)] cosx cos y = 1/2 [cos(x - y) + cos(x + y)] sinx cos y = 1/2 [sin(x + y) + sin(x - y)] cos x sin y = 1/2 [sin(x + y) - sin(x - y)]
even function
symmetric with respect to the y-axis
(sin x ) / (cos x)
tan x
Quotient Identities
tan x = (sin x)/(cos x) cot x = (cos x)/(sin x)
domain of a function
the set of all possible values of x for a function
range of a function
the set of all possible values of y for a function
e^(ln x)
x
quadratic formula
x = -b ± √(b² - 4ac)/2a
Sum of Cubes
x³ + y³ = (x + y)(x² - xy + y²)
Difference of Cubes
x³ - y³ = (x - y)(x² + xy + y²)
Inverse of y = ln (n)
y = eⁿ
inverse of y = eⁿ
y = ln (n)
point-slope
y-y₁ = m(x-x₁), where m is the slope and (x1,y1) is the point the line is passing through.
slope-intercept
y=mx+b
Product and Quotient of Two Complex Numbers
z1z2 = r1r2[cos(θ1 + θ2) + i*sin(θ1 +θ2)] Product z1/z2 = r1/r2[cos(θ1 - θ2) + i*sin(θ1 - θ2)] Quotient