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MTH167 Precalculus with Trigonometry Formula Sheet
Welcome to the comprehensive formula sheet for MTH167 Precalculus with Trigonometry. This page collects essential mathematical formulas and concepts that students need to master throughout the course. Use this resource as a quick reference when working through problems and preparing for exams.
1. Algebra Fundamentals
a(b + c) = ab + ac (Distributive Property)
(a + b) = a + 2ab + b (Perfect Square)
(a - b) = a - 2ab + b (Perfect Square)
(a + b)(a - b) = a - b (Difference of Squares)
(a + b)(c + d) = ac + ad + bc + bd (FOIL Method)
(a + b)(a - ab + b) = a + b (Sum of Cubes)
(a - b)(a + ab + b) = a - b (Difference of Cubes)
2. Exponents and Radicals
| Property | Formula |
|---|---|
| Product Rule | am an = am+n |
| Quotient Rule | am an = am-n |
| Power Rule | (am)n = amn |
| Power of a Product | (ab)n = an bn |
| Power of a Quotient | (a/b)n = an/bn |
| Zero Exponent | a0 = 1 (a 0) |
| Negative Exponent | a-n = 1/an |
| Fractional Exponent | am/n = (a1/n)m = nam |
Remember: na nb = n(ab)
3. Quadratic Functions and Equations
Standard Form
f(x) = ax + bx + c, a 0
Vertex Form
f(x) = a(x - h) + k, where (h, k) is the vertex
Factored Form
f(x) = a(x - x)(x - x)
Quadratic Formula
x = [-b (b - 4ac)]/(2a)
Axis of Symmetry
x = -b/(2a)
Vertex of a Parabola
h = -b/(2a), k = f(h) = -/(4a), where = b - 4ac
The discriminant determines the nature of the roots:
- > 0: Two distinct real roots
- = 0: One real root (repeated)
- < 0: Two complex conjugate roots
4. Polynomial Functions
Remainder Theorem
If polynomial P(x) is divided by (x - c), the remainder is P(c)
Factor Theorem
(x - c) is a factor of P(x) if and only if P(c) = 0
Rational Root Theorem
If a polynomial has integer coefficients, every rational zero has the form p/q, where p divides the constant term and q divides the leading coefficient
Fundamental Theorem of Algebra
A polynomial of degree n has exactly n complex roots (counting multiplicities)
Descartes' Rule of Signs
The number of positive real zeros of P(x) equals the number of sign changes in the coefficients of P(x) or is less by an even integer
5. Exponential and Logarithmic Functions
Exponential Growth/Decay
A = P(1 + r)t (Compound Interest)
A = Pert (Continuous Compounding)
Logarithmic Properties
| Property | Formula |
|---|---|
| Product Rule | loga(xy) = loga(x) + loga(y) |
| Quotient Rule | loga(x/y) = loga(x) - loga(y) |
| Power Rule | loga(xr) = rloga(x) |
| Change of Base | loga(x) = logb(x)/logb(a) |
| Inverse Properties | loga(ax) = x and aloga(x) = x |
Natural Logarithm and Exponential
ln(ex) = x and eln(x) = x
6. Trigonometric Functions
Basic Trigonometric Functions
| Function | Definition (in Right Triangle) | Definition (in Unit Circle) |
|---|---|---|
| Sine | sin() = opposite/hypotenuse | sin() = y-coordinate |
| Cosine | cos() = adjacent/hypotenuse | cos() = x-coordinate |
| Tangent | tan() = opposite/adjacent | tan() = sin()/cos() |
| Cosecant | csc() = hypotenuse/opposite | csc() = 1/sin() |
| Secant | sec() = hypotenuse/adjacent | sec() = 1/cos() |
| Cotangent | cot() = adjacent/opposite | cot() = cos()/sin() |
Pythagorean Identities
sin() + cos() = 1
1 + tan() = sec()
1 + cot() = csc()
Reciprocal Identities
sin() = 1/csc()
cos() = 1/sec()
tan() = 1/cot()
Quotient Identities
tan() = sin()/cos()
cot() = cos()/sin()
Even-Odd Identities
sin(-) = -sin() (Odd)
cos(-) = cos() (Even)
tan(-) = -tan() (Odd)
Cofunction Identities
sin(90 - ) = cos()
cos(90 - ) = sin()
tan(90 - ) = cot()
7. Trigonometric Identities
Sum and Difference Formulas
sin( + ) = sin()cos() + cos()sin()
sin( - ) = sin()cos() - cos()sin()
cos( + ) = cos()cos() - sin()sin()
cos( - ) = cos()cos() + sin()sin()
tan( + ) = [tan() + tan()]/[1 - tan()tan()]
tan( - ) = [tan() - tan()]/[1 + tan()tan()]
Double Angle Formulas
sin(2) = 2sin()cos()
cos(2) = cos() - sin() = 2cos() - 1 = 1 - 2sin()
tan(2) = 2tan()/[1 - tan()]
Half Angle Formulas
sin(/2) = [(1 - cos())/2]
cos(/2) = [(1 + cos())/2]
tan(/2) = sin()/[1 + cos()] = [1 - cos()]/sin()
Product-to-Sum Formulas
sin()cos() = [sin(+) + sin(-)]/2
cos()cos() = [cos(+) + cos(-)]/2
sin()sin() = [cos(-) - cos(+)]/2
Sum-to-Product Formulas
sin() + sin() = 2sin[(+)/2]cos[(-)/2]
sin() - sin() = 2cos[(+)/2]sin[(-)/2]
cos() + cos() = 2cos[(+)/2]cos[(-)/2]
cos() - cos() = -2sin[(+)/2]sin[(-)/2]
8. Inverse Trigonometric Functions
Domin Ranges
| Function | Domain | Range |
|---|---|---|
| arcsin(x) or sin(x) | [-1, 1] | [-/2, /2] |
| arccos(x) or cos(x) | [-1, 1] | [0, ] |
| arctan(x) or tan(x) | (-, ) | (-/2, /2) |
Inverse Properties
arcsin(sin()) = for -/2 /2
arccos(cos()) = for 0
arctan(tan()) = for -/2 < < /2
9. Laws of Sines and Cosines
Law of Sines
a/sin() = b/sin() = c/sin()
Law of Cosines
a = b + c - 2bccos()
b = a + c - 2accos()
c = a + b - 2abcos()
Area of a Triangle
Area = (1/2)bcsin() = (1/2)acsin() = (1/2)absin()
Area = [s(s-a)(s-b)(s-c)], where s = (a+b+c)/2 (Heron's Formula)
10. Analytic Geometry
Distance Formula
d = [(x - x) + (y - y)]
Midpoint Formula
M = ((x + x)/2, (y + y)/2)
Slope Formula
m = (y - y)/(x - x)
Equation of a Line
Point-Slope Form: y - y = m(x - x)
Slope-Intercept Form: y = mx + b
Standard Form: Ax + By = C
Parabola (Vertical)
(x - h) = 4p(y - k), vertex at (h, k)
Parabola (Horizontal)
(y - k) = 4p(x - h), vertex at (h, k)
Circle
(x - h) + (y - k) = r, center at (h, k), radius r
Ellipse
(x-h)/a + (y-k)/b = 1, center at (h, k), horizontal major axis
(x-h)/b + (y-k)/a = 1, center at (h, k), vertical major axis
Hyperbola
(x-h)/a - (y-k)/b = 1, center at (h, k), horizontal transverse axis
(y-k)/a - (x-h)/b = 1, center at (h, k), vertical transverse axis
11. Limits and Introduction to Calculus
Limit Properties
lim[f(x) g(x)] = lim f(x) lim g(x)
lim[kf(x)] = klim f(x)
lim[f(x)g(x)] = lim f(x)lim g(x)
lim[f(x)/g(x)] = lim f(x)/lim g(x), if lim g(x) 0
Special Limits
lim(x0) sin(x)/x = 1
lim(x0) (1 - cos(x))/x = 0
lim(x) (1 + 1/x)x = e
Continuity
A function f(x) is continuous at x = a if:
- f(a) is defined
- lim(xa) f(x) exists
- lim(xa) f(x) = f(a)
12. Sequences and Series
Arithmetic Sequences
Arithmetic Sequence: a, a+d, a+2d, ...
n-th term: a = a + (n-1)d
Sum of n terms: S = n/2(a + a) = n/2[2a + (n-1)d]
Geometric Sequences
Geometric Sequence: a, ar, ar, ...
n-th term: a = ar(n-1)
Sum of n terms: S = a(1-rn)/(1-r), r 1
Sum of infinite series: S = a/(1-r), |r| < 1
Fibonacci Sequence
F = 0, F = 1, and F = F + F for n 2
Binomial Theorem
(a + b)n = C(n,0)anb0 + C(n,1)an-1b1 + ... + C(n,n)a0bn
where C(n,k) = n!/(k!(n-k)!) (Binomial Coefficient)
13. Probability and Counting Principles
Counting Principles
Addition Principle: If event A can occur in m ways and event B can occur in n ways, then A or B can occur in m + n ways (if A and B are mutually exclusive)
Multiplication Principle: If event A can occur in m ways and after A occurs, event B can occur in n ways, then A and B can occur in m n ways
Permutations
P(n,r) = n!/(n-r)! (Number of ways to arrange r objects from a set of n distinct objects)
Combinations
C(n,r) = n!/(r!(n-r)!) (Number of ways to select r objects from a set of n distinct objects, order not important)
Basic Probability
P(A) = Number of favorable outcomes/Total number of possible outcomes
P(A B) = P(A) + P(B) - P(A B) (Addition Rule)
P(A B) = P(A|B) P(B) (Multiplication Rule)
P(A|B) = P(A B)/P(B) (Conditional Probability)
This formula sheet covers the core concepts of MTH167 Precalculus with Trigonometry. Regular practice with these formulas will strengthen your mathematical foundation and prepare you for advanced calculus courses. Remember to understand the concepts behind the formulas rather than simply memorizing them.
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