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