Precalculus Final Exam

∏/6

30 degrees

coordinate: (√3/2, 1/2)

∏/4

45 degrees

coordinate: (1/√2, 1/√2)

∏/3

60 degrees

coordinate: (1/2, √3/2)

∏/2

90 degrees

coordinate: (0, 1)

∏

180 degrees

coordinate: (-1, 0)

3∏/2

270 degrees

coordinate: (0, -1)

2∏

360 degrees

coordinate: (1, 0)

Trigonometric Functions in Quadrants:

Q1: sin, cos, and tan are positive

Q2: sin is positive, cos and tan are negative

Q3: sin and cos are negative, tan is positive

Q4: cos is positive, sin and tan are negative

Reciprocal Identities

sin x = 1/csc x

cos x = 1/sec x

tan x = 1/cot x

Quotient Identities

tan x = sin x/cos x

cot x = cos x/sin x

Odd Identities

sin (-x) = -sin x

tan (-x) = -tan x

Even Identities

cos (-x) = cos x

Periodicity

sin (x + 2∏) = sin x

cos (x + 2∏) = cos x

tan (x + ∏) = tan x

Compliments

sin x = cos (∏/2 - x)

tan x = cot (∏/2 - x)

sec x = csc (∏/2 - x)

Sum and Difference Formulas for Sine

sin (a + b) = (sin a)(cos b) + (cos a)(sin b)

sin (a - b) = (sin a)(cos b) - (cos a)(sin b)

Sum and Difference Formulas for Cosine

cos (a + b) = (cos a)(cos b) - (sin a)(sin b)

cos (a - b) = (cos a)(cos b) + (sin a)(sin b)

Double Angle Formulas

sin 2x = 2(sin x)(cos x)

cos 2x = cos^2 x - sin^2 x = 2cos^2 x - 1 = 1 - 2sin^2 x

tan 2x = (2tan x)/(1 -tan^2 x)

Half Angle Formulas

sin x/2 = ±√(1 - cos x)/2

cos x/2 = ±√(1 + cos x)/2

tan x/2 = (1 - cos x)/sin x = sin x/(1 + cos x)

Power Reducing Formulas

sin^2 x = (1 - cos 2x)/2

cos^2 x = (1 + cos 2x)/2

tan^2 x = (1 - cos 2x)/(1 + cos 2x)

Law of Sines

a/sin A = b/sin B = c/sin C

Law of Cosines

c^2 = a^2 + b^2 - 2ab(cosC)

f(x) = sin x

Domain: (-∞, ∞)

Range: [-1, 1]

Period: 2∏

Amplitude: 1

f(x) = cos x

Domain: (-∞, ∞)

Range: [-1, 1]

Period: 2∏

Amplitude: 1

f(x) = tan x

Domain: x ≠ ∏/2 + ∏n

Range: (-∞, ∞)

Period: ∏

Asymptotes: x = ∏/2 + ∏n

f(x) = cot x

Domain: x ≠∏n

Range: (-∞, ∞)

Period: ∏

Asymptotes: x = ∏n

f(x) = csc x

Domain: x ≠ ∏n

Range: (-∞, -1]U[1, ∞)

Period: ∏

Asymptotes: x = ∏n

f(x) = sec x

Domain: x ≠ ∏/2 + ∏n

Range: (-∞, -1]U[1, ∞)

Period: ∏

Asymptotes: x = ∏/2 + ∏n

f(x) = arcsin x

y = arcsin x ⟺ x = sin y

Domain: [-1, 1]

Range: [-∏/2, ∏/2]

f(x) = arccos x

y = arccos x ⟺ x = cos y

Domain: [-1, 1]

Range: [0, ∏]

f(x) = arctan x

y = arctan x ⟺ x = tan y

Domain: (-∞, ∞)

Range: [-∏/2, ∏/2]

One-to-One Function

-For every input (x-value), there is one and only one output (y-value)

-Graph must pass vertical line test

End Behavior of Function

End behavior is always determined by the highest power term (odd or even) and the leading coefficient (positive or negative)

Turning Points

-There are at most n-1 turning points

-Actual number of turning points = n-1, n-3, n-5, etc.

Multiplicity

Multiplicity determines how function behaves as it approaches an x-intercept (even multiplicity bounces, odd multiplicity crosses axis)

Horizontal/Oblique Asymptotes of Rational Functions

-If degree of denominator > degree of numerator, function has horizontal asymptote y=0

-If degree of denominator = degree of numerator, function has horizontal asymptote = ratio of leading coefficients

-If degree of denominator < degree of numerator, function has oblique asymptote = quotient of polynomials

Vertical Asymptotes of Rational Functions

Rational function has vertical asymptotes at values where denominator of simplified function = 0

Holes

Rational function has holes when terms in denominator are cancelled out by terms in numerator

Rational Zeros Test

Possible rational zeroes = ±(p/q), where p = constant and q = leading coefficient

Exponential Function

Domain: (-∞, ∞)

Range: (0, ∞)

Asymptote: y=0

Intercept: (0, 1)

Always increasing

Logarithmic Function

Domain: (0, ∞)

Range: (-∞, ∞)

Asymptote: x=0

Intercept: (1, 0)

Always increasing

Properties of Logarithms

log(mn) = log(m) + log(n)

log(m/n) = log(m) - log(n)

log(m)^n = n log(m)

Properties of Exponents

a^mn = (a^m)^n

a^(m + n) = (a^m)(a^n)

Distance Formula

d = √(x1 - x2)^2 - (y1 - y2)^2

Midpoint Formula

midpoint = (x1 + x2)/2, (y1 + y2)/2

Parabolas

Opens up: (x - h)^2 = 4(a)(y - k)

Opens down: (x - h)^2 = -4(a)(y - k)

Opens right: (y - k)^2 = 4(a)(x - h)

Opens left: (y - k)^2 = -4(a)(x - h)

Ellipses

Major axis is horizontal: (x - h)^2/a^2 + (y - k)^2/b^2 = 1

Major axis is vertical: (x - h)^2/b^2 + (y - k)^2/a^2 = 1

a^2 = b^2 + c^2

Hyperbolas

Transverse axis is horizontal: (x - h)^2/a^2 - (y - k)^2/b^2 = 1

Transverse axis is vertical: (y - k)^2/a^2 - (x - h)^2/b^2 = 1

c^2 = a^2 + b^2

Classifying Conics from General Form Equation

Ax^2 + By^2 + Cx + Dy + F = 1

If A = C, conic is a circle

If AC = 0, conic is a parabola

If AC > 0, conic is an ellipse

If AC < 0, conic is a hyperbola

Arithmetic Sequences

a(n) = a(1) - (n - 1)d

∑ = n/2 [a(1) + a(n)]

Geometric Sequences

a(n) = a(1) x r^(n - 1)

∑ = a(1) x [(1 - r^n)/(1 - r)]

Convergence of a Geometric Series

∞ ∑ = a(1)/ (1 -r), provided that |r| < 0

Formula for Annuity

A = P x [(1 + i)^n -1]/i

i = interest/payments per year

n = years

Binomial Theorem

(x + a)^n = (n 0)x^n + (n 1)x^(n-1)a + ... + (n j)x^(n-j)a^j + ... + (n n)a^n

P(A ∩ B)

P(A and B) = all outcomes both in A and B (the overlap)

P(A U B)

P(A or B) = all outcomes in at least one of A or B

Complement of an Event

A^C = the outcomes NOT in an event

Conditional Probability

Probability that A happens given that B has already occurred

P(A|B) = P(A and B)/P(B)

Average Rate of Change

slope of secant line (msec) = [f(x1) - f(x2)] / (x1 - x2)

Limits

Limit exists only if left hand limit = right hand limit

Continuity

Function is continuous at x = c if:

1. f(c) is defined

2. limit as f → c exists

3. limit as f → c = f(c)

Mathematical Induction

Condition 1: The statement is true for the natural number 1

Condition 2: If the statement is true for some natural number k, then it is also true for the next natural number k+1