Algebra and Trigonometry 2e PDF Download – Jay Abramson (OpenStax) BS Math

Algebra and Trigonometry 2e is OpenStax’s combined text for a university algebra-and-trigonometry course — functions, polynomial and rational functions, exponential and logarithmic functions, and the full trigonometry sequence (unit circle, identities, applications) in one book. Published December 21, 2021 by OpenStax at Rice University, it is built around worked examples in a two-part Solution/Analysis format, and every section ends with “Try It” practice problems that mirror the worked examples directly.

For BS Mathematics students, this single book does the job that two separate semester courses — College Algebra and Trigonometry — often split between them: it builds the algebraic manipulation, function, and graphing skills a Calculus sequence assumes, then covers the unit-circle and identity work trigonometry itself demands. Students preparing for a placement or entry test, or revising before Calculus I, get both halves of that preparation from one consistently structured text.

Book Overview

Chapter List

Chapter 1: Prerequisites

Difficulty: Easy · Key topics: real numbers, order of operations, properties of real numbers, exponents and scientific notation, radicals and rational exponents, polynomials, factoring, rational expressions

Chapter 1 rebuilds the algebra toolkit the rest of the book assumes, starting from how numbers themselves are classified. It reviews the real number system — rational versus irrational, and the order of operations and properties (commutative, associative, distributive, identity, inverse) that govern how expressions are manipulated — before moving into the mechanics: exponent rules and scientific notation, radicals and rational exponents, then polynomial arithmetic, factoring, and rational expressions. Each section builds directly on the one before it, ending with the tools needed to simplify a complex rational expression.

Key Points:

• A number is rational if it can be written as a fraction or a terminating/repeating decimal; otherwise it is irrational, and together they form the real numbers.
• The real numbers obey the commutative, associative, distributive, identity, and inverse properties under addition and multiplication.
• Products, quotients, and powers of exponential expressions with the same base are simplified by adding, subtracting, or multiplying exponents; a zero exponent equals 1 and a negative exponent gives a reciprocal.
• The principal nth root of a can be rewritten as the rational exponent a1/n, and the exponent rules apply the same way to rational exponents.
• A polynomial’s degree is its highest power; the leading term and leading coefficient come from that term.
• Checking for a greatest common factor (GCF) should always be the first step in factoring — trinomials, grouping, perfect-square, and sum/difference-of-cubes patterns come after.
• Rational expressions are simplified by cancelling common factors, and added or subtracted only after finding a common denominator.

Memory Tip: Before Chapter 2, make sure GCF-first is automatic: given any expression to factor, your first move — before trying grouping or a special pattern — should always be “is there a common factor in every term?”

Common Mistake: Treating √(a²+b²) as a+b. Radicals do not distribute over addition — √(9+16)=√25=5, not √9+√16=3+4=7. The same error shows up later with rational exponents on binomials.

Important Questions:

• Is every rational number also a real number, and is every real number rational? Yes to the first — rational numbers are a subset of the reals; no to the second — irrational numbers (like √2) are real but not rational.
• Factor 6x² + 7x − 3 completely. (3x − 1)(2x + 3), found by identifying factors of 6×(−3)=−18 that sum to 7 (9 and −2) and factoring by grouping.

Chapter 2: Equations and Inequalities

Difficulty: Easy · Medium · Key topics: rectangular coordinates and graphs, linear equations, models and applications, complex numbers, quadratic equations, other equation types, linear and absolute value inequalities

Chapter 2 moves from simplifying expressions to solving for unknowns. It opens by plotting points and equations on the Cartesian plane and deriving the distance and midpoint formulas, then works through linear equations in one variable, including rational equations that need a common denominator cleared first. A section on setting up word problems as equations follows, before the chapter introduces complex numbers to handle negative square roots, brings in the full toolkit for quadratic equations (factoring, the square root property, completing the square, and the quadratic formula), extends to radical, absolute-value, and quadratic-form equations, and closes with linear and absolute-value inequalities in interval notation.

Key Points:

• The distance formula, √((x₂−x₁)²+(y₂−y₁)²), comes directly from the Pythagorean Theorem applied to a coordinate plane.
• A rational equation is solved by clearing denominators with the LCD, but every solution must be checked back in the original equation to rule out a zero denominator.
• The square root of a negative number is written as a multiple of i, and the powers of i repeat in a cycle of four (i, −1, −i, 1).
• The discriminant (b²−4ac) tells you the nature of a quadratic’s roots — real or complex, rational or irrational — before you solve it.
• Solving a radical equation means isolating the radical and raising both sides to a matching power — and always checking for extraneous solutions afterward.
• When solving an inequality, multiplying or dividing both sides by a negative number is the one step that reverses the inequality symbol.

Memory Tip: Whenever a step could introduce a “fake” solution — squaring both sides, clearing a variable denominator, or taking an even root — treat the answer as a candidate, not a solution, until it is checked in the original equation.

Common Mistake: Forgetting to reverse an inequality sign when multiplying or dividing by a negative number — for example, turning −2x < 6 into x < −3 instead of the correct x > −3.

Important Questions:

• Solve x² − 4x + 13 = 0 using the quadratic formula. The discriminant is 16−52=−36, so the roots are complex: x = 2 ± 3i.
• What is i⁴⁵ equal to, and why? i (since powers of i repeat every 4: i¹=i, i²=−1, i³=−i, i⁴=1, and 45 mod 4 = 1, so i⁴⁵=i¹=i).

Chapter 3: Functions

Difficulty: Medium · Key topics: function notation, domain and range, rates of change, local extrema, composition of functions, transformations, absolute value functions, inverse functions

Chapter 3 is the hinge the rest of the book turns on. It formally defines a function as a relation where every input has exactly one output, introduces f(x) notation, and works through evaluating, tabulating, and graphing functions, including the vertical-line and horizontal-line tests for functions and one-to-one functions. From there it covers domain and range (including piecewise functions), average rate of change and identifying local maxima/minima from a graph, composing two functions into one, the full set of graph transformations (shifts, reflections, stretches, compressions), the absolute value function specifically, and finally how to find and verify an inverse function algebraically and graphically.

Key Points:

• A relation is a function only if every input maps to exactly one output; the vertical line test checks this on a graph.
• A function is one-to-one only if it also passes the horizontal line test — this is the condition required for an inverse function to exist.
• Average rate of change over an interval is computed from the function’s values at the interval’s endpoints, using only that beginning and ending data.
• A composite function’s domain is limited to inputs in the inner function’s domain whose outputs also lie in the outer function’s domain.
• Vertical shifts add a constant to the output; horizontal shifts add a constant to the input — and the two types of shift can be combined.
• A function is even if f(−x)=f(x) (symmetric about the y-axis) or odd if f(−x)=−f(x) (symmetric about the origin); most functions are neither.
• To find an inverse algebraically: solve y=f(x) for x in terms of y, then swap the labels x and y.

Memory Tip: “Vertical line = is it a function; horizontal line = does it have an inverse.” Every transformation and inverse question in this chapter reduces back to one of those two tests.

Common Mistake: Assuming every function has an inverse function. Only a one-to-one function does — which is exactly why arcsin, arccos, and arctan (Chapter 8) are defined only on restricted domains of sine, cosine, and tangent.

Important Questions:

• If f(x) = 3x − 5 and g(x) = x², what is (f ∘ g)(2)? g(2)=4, then f(4)=3(4)−5=7, so (f ∘ g)(2)=7.
• Is f(x) = x³ even, odd, or neither, and how do you know? Odd — f(−x)=(−x)³=−x³=−f(x) for every x, so its graph is symmetric about the origin.

Chapter 4: Linear Functions

Difficulty: Easy · Medium · Key topics: linear function representations, slope, modeling with linear functions, scatter plots, line of best fit, correlation coefficient, regression

Chapter 4 applies Chapter 3’s function ideas to the simplest, most widely used function type. It covers the four ways a linear function can be represented (words, notation, table, graph), how slope is calculated and read off a graph, writing a line’s equation from a graph, from a slope and point, or from two points, and identifying parallel and perpendicular lines from their slopes. It then applies linear functions to real word problems, and closes by fitting a line to scattered real-world data — interpolation versus extrapolation, the correlation coefficient, and the least-squares regression line.

Key Points:

• Slope is a rate of change: the difference in y-values divided by the difference in x-values between any two points on the line.
• An increasing linear function has positive slope; a decreasing one has negative slope; a constant function graphs as a horizontal line with slope 0.
• Parallel lines share the same slope; perpendicular lines have slopes that are negative reciprocals of each other (unless one is vertical and the other horizontal).
• Horizontal lines are written f(x)=b; vertical lines are written x=b and have an undefined slope.
• Interpolation predicts values inside the range of the collected data; extrapolation predicts values outside it — and is far less reliable.
• The correlation coefficient r indicates how strong a linear relationship the data actually has, before you trust a regression line built from it.

Memory Tip: Slope-intercept form y=mx+b: m is what you multiply x by (the rate), b is what’s there before x does anything (the starting value, at x=0).

Common Mistake: Computing slope as (x₂−x₁)/(y₂−y₁) — upside down. Slope is rise over run: change in y divided by change in x, always.

Important Questions:

• Find the equation of the line through (2, 5) and (4, 11). Slope = (11−5)/(4−2)=3; using point-slope with (2,5): y−5=3(x−2), which simplifies to y=3x−1.
• What does a correlation coefficient close to r=0 tell you about a scatter plot? That there is little to no linear relationship between the two variables — a regression line fit to that data would not reliably predict new values.

Chapter 5: Polynomial and Rational Functions

Difficulty: Medium · Hard · Key topics: quadratic functions and the vertex, power and polynomial functions, end behavior, graphing polynomials, polynomial division, zeros of polynomials, rational functions and asymptotes, inverse and radical functions, variation

Chapter 5 is one of the book’s longest, covering every function family built from powers of x. It opens with quadratic functions in general and vertex form, then generalizes to power and polynomial functions and their end behavior (determined entirely by the leading term). Graphing polynomial functions combines zeros, multiplicity, and end behavior into one procedure. Polynomial long division and synthetic division follow, feeding directly into finding a polynomial’s zeros with the Remainder Theorem, Factor Theorem, and Rational Zero Theorem. The chapter then turns to rational functions — vertical and horizontal asymptotes, removable discontinuities — before closing with inverses of polynomial/radical functions and direct, inverse, and joint variation.

Key Points:

• A quadratic function’s minimum or maximum value is exactly the y-value of its vertex, found from either general or vertex form.
• A polynomial function’s end behavior is decided entirely by its leading term — the degree and sign of the leading coefficient, nothing else.
• A polynomial of degree n has at most n x-intercepts and at most n−1 turning points.
• The graph crosses the x-axis at a zero of odd multiplicity, but only touches it at a zero of even multiplicity.
• By the Rational Zero Theorem, every rational zero of a polynomial with integer coefficients equals a factor of the constant term divided by a factor of the leading coefficient.
• Vertical asymptotes of a rational function occur where the denominator is zero and the numerator is not; a removable discontinuity occurs where both are zero at once.
• A rational function’s end behavior mirrors the ratio of its numerator’s and denominator’s leading terms.

Memory Tip: “Multiplicity odd = crosses, multiplicity even = bounces.” Sketch every polynomial zero by asking only that one question before drawing the curve through it.

Common Mistake: Assuming a rational function has a vertical asymptote at every zero of the denominator. If the numerator is also zero there, it is a removable discontinuity (a hole), not an asymptote — check both before graphing.

Important Questions:

• What is the end behavior of f(x) = −2x⁴ + 3x² − 1? Even degree with a negative leading coefficient, so both ends fall: as x→±∞, f(x)→−∞.
• Find the vertical asymptotes of f(x) = (x²−4)/(x²−x−6). Factor: (x−2)(x+2)/((x−3)(x+2)) — the (x+2) cancels, leaving a hole at x=−2, and the only vertical asymptote is at x=3.

Chapter 6: Exponential and Logarithm Functions

Difficulty: Medium · Hard · Key topics: exponential functions, compound interest, the number e, graphs and transformations, logarithmic functions, logarithm graphs, logarithm properties, solving exponential and logarithmic equations, exponential/logarithmic models, fitting exponential models to data

Chapter 6 introduces the two function families that behave unlike anything in the book so far. It defines exponential functions, builds models from a growth rate or from data points, and introduces the compound interest formula and the constant e for continuous growth. Graphs of exponential functions and their transformations follow the same shift/stretch/reflection framework as Chapter 3. The chapter then defines logarithmic functions as the inverse of exponential functions, covers their graphs and vertical asymptotes, the product/quotient/power rules of logarithms and the change-of-base formula, solving exponential and logarithmic equations (including with base e), and closes with real-world growth/decay models — including carbon-14 dating and Newton’s Law of Cooling — and fitting exponential, logarithmic, or logistic regression to data.

Key Points:

• An exponential function f(x)=bx has y-intercept (0,1), domain all reals, range (0,∞), and horizontal asymptote y=0.
• If b>1 the function increases; if 0<b<1 it decreases — the graph’s shape depends entirely on which side of 1 the base falls.
• The constant e≈2.718282 is the base used for continuous growth and decay models, A=A₀ekt.
• A logarithmic equation and an exponential equation are two ways of writing the same relationship — logb(x)=y is equivalent to by=x.
• The product, quotient, and power rules let you expand or condense a logarithm of a product, quotient, or power into sums, differences, or multiples of simpler logarithms.
• The change-of-base formula converts any logarithm into a quotient of natural or common logs so it can be evaluated on a calculator.
• Logistic growth models level off toward an upper limit over time, unlike plain exponential growth, which is why they fit things like population growth or the spread of a rumor better.

Memory Tip: Logarithms and exponentials undo each other exactly the way multiplication and division do — whenever an equation has a variable trapped inside a logarithm or an exponent, apply the inverse operation to both sides to free it.

Common Mistake: Writing log(x+y) as log(x)+log(y). There is no rule for the log of a sum — only for the log of a product, quotient, or power. log(x)+log(y) equals log(xy), not log(x+y).

Important Questions:

• Solve 3^2x−1 = 27 for x. Rewrite 27 as 3³, so 2x−1=3, giving x=2.
• Expand log₃(9x²/y) fully using logarithm properties. log₃(9)+log₃(x²)−log₃(y) = 2+2log₃(x)−log₃(y), using the product rule, quotient rule, and power rule in that order.

Chapter 7: The Unit Circle: Sine and Cosine Functions

Difficulty: Medium · Key topics: angles, degrees and radians, coterminal angles, arc length and sector area, angular and linear speed, right-triangle trigonometry, the unit circle, reference angles, the six trigonometric functions

Chapter 7 opens the trigonometry half of the book. It defines an angle as formed by rotating a terminal ray from an initial side, standard position, and the conversion between degrees and radians, then covers coterminal angles, arc length, sector area, and the angular/linear speed of an object moving in a circle. Right-triangle trigonometry defines sine, cosine, and tangent as side-length ratios and introduces the cofunction identities. The unit circle (radius 1, centered at the origin) then gives sine and cosine a cleaner definition — the y- and x-coordinates of a point on the circle — using reference angles to extend known values into every quadrant, before the chapter defines the remaining four trigonometric functions (tangent, secant, cosecant, cotangent) as ratios and reciprocals of sine and cosine.

Key Points:

• To convert between degrees and radians, use θ/180 = θR/π — multiply by π/180 to go from degrees to radians, or 180/π to go the other way.
• On the unit circle, cos(t) is the x-coordinate and sin(t) is the y-coordinate of the point reached by an arc of length t.
• The Pythagorean Identity sin²(t)+cos²(t)=1 lets you find sine from cosine or cosine from sine, and holds for every real t.
• A reference angle is the acute angle between the terminal side and the x-axis; sine and cosine of any angle share the same absolute value as their reference angle, with sign set by the quadrant.
• Tangent is the ratio y/x on the unit circle; secant, cosecant, and cotangent are the reciprocals of cosine, sine, and tangent respectively.
• Cosine and secant are even functions; sine, tangent, cosecant, and cotangent are odd — this determines how each behaves under a sign change of the input.

Memory Tip: “All Students Take Calculus” — going counterclockwise from Quadrant I, that names which functions are positive in each quadrant: All (QI), Sine (QII), Tangent (QIII), Cosine (QIV).

Common Mistake: Mixing up which coordinate is sine and which is cosine on the unit circle. Cosine is x (think “c comes before s, x comes before y”) and sine is y — reversing them flips every quadrant-sign answer that follows.

Important Questions:

• Convert 210° to radians. 210 × (π/180) = 7π/6.
• Find the exact value of cos(5π/3) using a reference angle. The reference angle is π/3 (60°), in Quadrant IV where cosine is positive, so cos(5π/3) = cos(π/3) = 1/2.

Chapter 8: Periodic Functions

Difficulty: Medium · Key topics: graphing sine and cosine, amplitude, period, phase shift, vertical shift, graphing tangent/secant/cosecant/cotangent, inverse trigonometric functions

Chapter 8 turns the unit-circle definitions from Chapter 7 into graphs, and asks what it means to undo a trigonometric function. It shows that sine and cosine graphs are odd/even respectively, with a base period of 2π, and walks through the general sinusoidal formula — how amplitude, period, phase shift, and vertical shift are each read straight off the equation and combined to graph or reconstruct a sinusoidal model from real data. It then graphs tangent (period π), secant, cosecant, and cotangent along with their asymptotes, before defining the inverse trigonometric functions on restricted domains and working through evaluating compositions like sin(cos⁻¹(x)) using reference triangles.

Key Points:

• In the general sinusoidal formula, the period is P=2π/|B| and the amplitude is |A| — stretching the graph vertically if |A|>1, compressing it if |A|<1.
• The phase shift of a sinusoidal function is C/B, and the vertical shift is D, from the general form.
• The tangent function has period π (half of sine and cosine’s), and the secant/cosecant functions keep sine/cosine’s period of 2π but are undefined wherever the reciprocal function is zero.
• An inverse function’s domain is the original function’s range and vice versa — but since none of the six trig functions are one-to-one on their full domain, each inverse is defined only on a restricted domain of the original.
• A calculator always returns an angle inside that restricted domain, even when other coterminal or equivalent angles also solve the original equation.
• Composing a trig function with an inverse trig function of a different type (e.g. sin(cos⁻¹(x))) is evaluated by drawing a reference right triangle from the inner inverse function’s ratio.

Memory Tip: Read a sinusoidal equation left to right as “stretch, squeeze-the-period, shift-sideways, shift-up-down” — A controls amplitude, B controls how fast it repeats, C/B slides it horizontally, D slides it vertically.

Common Mistake: Reading the period straight off B instead of computing 2π/|B|. For y=sin(3x), the period is 2π/3, not 3 — a larger B means a shorter period, not a longer one.

Important Questions:

• What are the amplitude and period of y = 4sin(2x)? Amplitude = |4| = 4; period = 2π/|2| = π.
• Evaluate sin(cos⁻¹(3/5)) using a reference triangle. cos⁻¹(3/5) is the angle in a right triangle with adjacent=3, hypotenuse=5, so opposite=4 (3-4-5 triple); sin of that angle is opposite/hypotenuse = 4/5.

Chapter 9: Trigonometric Identities and Equations

Difficulty: Hard · Key topics: verifying identities, sum and difference identities, double-angle and half-angle formulas, reduction formulas, sum-to-product and product-to-sum formulas, solving trigonometric equations

Chapter 9 is one of the two hardest chapters in the book, because it asks students to manipulate trigonometric expressions algebraically rather than just evaluate them. It opens with strategies for verifying an identity — usually by simplifying the more complex side of the equation using the fundamental identities from Chapter 7 — then derives the sum and difference identities for sine, cosine, and tangent, and uses them to build the double-angle, reduction, and half-angle formulas. Product-to-sum and sum-to-product formulas follow from the same sum/difference identities. The chapter closes by applying all of this to actually solving trigonometric equations: algebraic techniques, the unit circle, substitution for quadratic-form equations, and using the identities themselves as solving tools.

Key Points:

• Verifying an identity means starting from the more complicated side of the equation and simplifying it — using fundamental identities — until it matches the other side exactly.
• The cosine sum/difference identities are cos(α±β) = cosαcosβ ∓ sinαsinβ; the sine versions swap which functions multiply which.
• Double-angle identities (e.g. sin(2θ)=2sinθcosθ) are derived directly from the sum formulas by setting the two angles equal.
• Half-angle formulas let you find trig values at half of a known angle, even when the original angle’s exact value is unknown.
• Product-to-sum and sum-to-product formulas rewrite a product of sines/cosines as a sum (or vice versa) — useful for simplifying an expression before integrating it in calculus.
• Solving a trig equation that “looks quadratic” (e.g. in sinθ) uses the same substitution, factoring, or quadratic-formula techniques as an ordinary algebraic quadratic.

Memory Tip: When stuck verifying an identity, convert everything to sine and cosine first — tangent, secant, cosecant, and cotangent are all built from sin and cos, and the identity usually becomes obvious once there are only two functions left to compare.

Common Mistake: Treating sin(A+B) as sin(A)+sin(B). Trigonometric functions do not distribute over addition — the correct sum identity is sin(A+B)=sinAcosB+cosAsinB, and skipping straight to sinA+sinB is one of the most common errors in this chapter.

Important Questions:

• Use a sum identity to find the exact value of cos(75°). 75°=45°+30°, so cos(75°)=cos45°cos30°−sin45°sin30° = (√2/2)(√3/2)−(√2/2)(1/2) = (√6−√2)/4.
• Solve 2sin²θ − sinθ − 1 = 0 for θ in [0, 2π). Factoring as a quadratic in sinθ: (2sinθ+1)(sinθ−1)=0, so sinθ=−1/2 or sinθ=1, giving θ = 7π/6, 11π/6, and π/2.

Chapter 10: Further Applications of Trigonometry

Difficulty: Hard · Key topics: Law of Sines, Law of Cosines, Heron’s formula, polar coordinates and graphs, polar form of complex numbers, parametric equations and graphs, vectors, dot product

Chapter 10 is the book’s longest and most varied chapter, extending trigonometry beyond right triangles into several genuinely different coordinate systems. It opens by solving oblique (non-right) triangles with the Law of Sines, including the ambiguous SSA case, then the Law of Cosines for SAS/SSS triangles and Heron’s formula for area from three known sides. Polar coordinates follow — plotting points, converting to and from rectangular form, and graphing polar equations including cardioids, limaçons, lemniscates, and rose curves, based on symmetry tests. The chapter then writes complex numbers in polar form for easier multiplication, division, and roots, introduces parametric equations (a curve defined by x(t) and y(t) sharing a parameter t) with projectile motion as a worked application, and closes with vectors — magnitude, direction, addition, scalar multiplication, unit vectors, and the dot product.

Key Points:

• The Law of Sines states the ratio of each angle’s sine to its opposite side’s length is the same for all three angles of any triangle.
• The SSA case (Law of Sines) is called the ambiguous case because it can yield zero, one, or two valid triangles from the same given information.
• The Law of Cosines (c²=a²+b²−2ab·cosC) is the generalized Pythagorean Theorem, needed whenever a triangle has no right angle and SAS or SSS information.
• To convert polar to rectangular coordinates, use x=rcosθ and y=rsinθ; to go the other way, use r=√(x²+y²) and tanθ=y/x.
• A complex number in polar form is written z=r(cosθ+isinθ), which makes multiplying (multiply moduli, add angles) and finding roots far easier than in rectangular form.
• Eliminating the parameter t from a set of parametric equations recovers the equivalent rectangular equation, usually by solving one equation for t and substituting into the other.
• Two vectors are equal only if they share both magnitude and direction; the dot product of ai+bj and ci+dj is ac+bd, and is used to find the angle between two vectors.

Memory Tip: Picking a Law: two angles and a side, or two sides and a non-included angle → Law of Sines. Two sides and the included angle, or all three sides → Law of Cosines. That one question decides the whole first half of the chapter.

Common Mistake: Forgetting to check the ambiguous case when solving an SSA triangle with the Law of Sines — reporting only one triangle when the given angle and sides actually allow two valid solutions (or none at all).

Important Questions:

• In triangle ABC, a=7, b=10, and angle A=40°. Use the Law of Sines to set up the equation for angle B. sinB/10 = sin40°/7, so sinB = 10sin40°/7 ≈ 0.918 — this must then be checked for the ambiguous case since it’s SSA.
• Convert the point (r,θ) = (4, π/3) from polar to rectangular coordinates. x = 4cos(π/3) = 4(1/2) = 2; y = 4sin(π/3) = 4(√3/2) = 2√3, so the rectangular point is (2, 2√3).

Chapter 11: Systems of Equations and Inequalities

Difficulty: Hard · Key topics: systems of linear equations (2 and 3 variables), nonlinear systems, systems of inequalities, partial fractions, matrices and matrix operations, Gaussian elimination, matrix inverses, Cramer’s Rule

Chapter 11 shifts from single equations to solving several at once. It covers solving 2-variable linear systems by graphing, substitution, and elimination, classifying systems as independent, dependent, or inconsistent, then extends the same elimination logic to 3-variable systems and geometrically interprets what an inconsistent or dependent 3-variable system means (parallel or coincident planes). Nonlinear systems (a line and a parabola, a circle and an ellipse) and systems of inequalities follow, then partial fraction decomposition for rewriting a rational expression as a sum of simpler fractions. The rest of the chapter is built around matrices: matrix arithmetic and multiplication, Gaussian elimination to row-echelon form, finding and using a matrix inverse to solve AX=B, and Cramer’s Rule using determinants.

Key Points:

• A 2-variable linear system is independent (one solution), dependent (infinite solutions — same line), or inconsistent (no solution — parallel lines), decided by what happens when you solve it.
• Matrix multiplication requires the first matrix’s number of columns to equal the second matrix’s number of rows — the “inner dimensions” must match.
• An augmented matrix bundles a system’s coefficients and constants together so row operations can solve the whole system at once.
• Gaussian elimination reduces an augmented matrix to row-echelon form, after which back-substitution finds every variable.
• A matrix has an inverse only if AA−1=A−1A=I; once found, a system AX=B is solved directly as X=A−1B.
• Cramer’s Rule solves for each variable as a ratio of determinants, replacing one column of the coefficient matrix with the constants column at a time.
• The determinant of a 2×2 matrix [[a,b],[c,d]] is ad−bc; if it equals 0, the matrix has no inverse.

Memory Tip: Choosing a solving method: two variables and a quick check → substitution or elimination by hand. Three or more variables, or asked for “the matrix method” → augmented matrix and Gaussian elimination. Need one variable’s value only, fast → Cramer’s Rule.

Common Mistake: Multiplying matrices entry-by-entry like addition (multiplying corresponding positions together). Matrix multiplication is row-times-column: each entry of the product comes from summing the products of a full row of the first matrix against a full column of the second.

Important Questions:

• Solve the system x+y=10, x−y=2 by elimination. Adding the two equations eliminates y: 2x=12, so x=6, and substituting back gives y=4.
• What does it mean if Gaussian elimination on a 3-variable system produces the row [0 0 0 | 5]? The system is inconsistent (no solution) — that row translates to 0=5, a contradiction, which geometrically means the three planes never share a common point.

Chapter 12: Analytic Geometry

Difficulty: Medium · Hard · Key topics: the ellipse, the hyperbola, the parabola, foci and directrices, rotation of axes, conic sections in polar coordinates

Chapter 12 studies the curves formed when a plane cuts through a cone — the conic sections. It defines an ellipse as the set of points whose distances to two foci sum to a constant, and works through writing its standard-form equation and reading off its vertices, co-vertices, and foci, both centered at the origin and shifted. The hyperbola gets the same treatment, defined by a constant difference of distances to two foci instead of a sum, including its asymptotes. The parabola is defined via a focus and directrix (equidistant from both), with four standard orientations depending on the sign of p. The chapter then shows how an xy cross-term in a conic’s general equation signals the conic has been rotated, and gives a rotation-of-axes method to eliminate that term, before closing with a unified polar-coordinate definition of all the conics using eccentricity.

Key Points:

• An ellipse is the set of points whose distances to two fixed foci sum to a constant; a hyperbola is the set whose distances to two foci have a constant difference.
• A parabola is the set of points equidistant from a fixed focus and a fixed line, the directrix — neither point lies on the other.
• The standard-form equation of a conic centered at the origin directly gives its vertices, co-vertices, and foci without needing to complete the square first.
• A hyperbola’s asymptotes are found directly from its standard-form equation and guide the shape of its two branches.
• A general conic equation Ax²+Bxy+Cy²+Dx+Ey+F=0 has been rotated about the origin whenever B≠0; rotation of axes removes that xy-term.
• Every conic can be defined by a single eccentricity value e: e<1 gives an ellipse, e=1 a parabola, and e>1 a hyperbola.

Memory Tip: Sum of distances to two foci = ellipse (think “e” for ellipse and “equal total”); difference of distances = hyperbola. That one word — sum vs. difference — is the entire distinction between the two definitions.

Common Mistake: Mixing up which denominator (a² or b²) goes under x² versus y² in an ellipse or hyperbola’s standard form, which flips whether the major/transverse axis is horizontal or vertical. Always check which variable’s denominator is larger (ellipse) or which term is positive (hyperbola) before sketching.

Important Questions:

• What are the foci of the ellipse x²/25 + y²/9 = 1? a²=25, b²=9, so c²=a²−b²=16, c=4; since the major axis is horizontal (larger denominator under x²), the foci are at (±4, 0).
• How can you tell from Ax²+Bxy+Cy²+Dx+Ey+F=0 whether a conic has been rotated? If B≠0 (there is an xy term), the conic’s axes are rotated relative to the coordinate axes, and rotation of axes is needed to identify its standard form.

Chapter 13: Sequences, Probability, and Counting Theory

Difficulty: Medium · Key topics: sequences and notation, arithmetic sequences, geometric sequences, series and summation notation, annuities, counting principles, permutations and combinations, the Binomial Theorem, probability

Chapter 13 closes the book with discrete mathematics topics that, unlike most of the rest of the text, don’t depend heavily on trigonometry. It defines sequences via explicit and recursive formulas, then specializes to arithmetic sequences (constant common difference) and geometric sequences (constant common ratio), each with its own recursive and explicit formula. Series — the sum of a sequence’s terms — get their own summation notation and formulas for arithmetic and geometric sums, including infinite geometric series and their application to annuities. The chapter then covers the Addition and Multiplication counting principles, permutations and combinations (order matters versus doesn’t), the Binomial Theorem for expanding (x+y)n without multiplying it out, and closes with basic probability, including the probability of a union of events and of a complement.

Key Points:

• An arithmetic sequence has a constant difference between consecutive terms; a geometric sequence has a constant ratio between them.
• The explicit formula for an arithmetic sequence is an=a₁+d(n−1); for a geometric sequence it is an=a₁rn−1.
• An infinite geometric series has a finite sum only when −1<r<1; that condition is what makes an annuity’s value calculable with a simple formula.
• A permutation counts orderings (order matters); a combination counts selections (order doesn’t) — the same set of objects gives a much larger permutation count than combination count.
• The Binomial Theorem expands (x+y)n using binomial coefficients C(n,r), avoiding the need to multiply the binomial out n times.
• Probability is always between 0 and 1, and the probability of the complement of an event is 1 minus the probability the event occurs.

Memory Tip: “Difference is arithmetic, ratio is geometric” — and for counting, ask “does the order I pick things in matter?” before reaching for a permutation or combination formula.

Common Mistake: Using a combination formula when order actually matters (or vice versa) — for example, counting how many ways to award 1st/2nd/3rd place (a permutation, P(n,r)) using the combination formula C(n,r), which silently divides out the distinct orderings and undercounts the answer.

Important Questions:

• Find the 10th term of the arithmetic sequence with a₁=4 and d=3. a₁₀ = 4 + 3(10−1) = 4+27 = 31.
• In how many ways can a committee of 3 be chosen from 8 people? Order doesn’t matter, so use combinations: C(8,3) = 8!/(3!5!) = 56.

Download Algebra and Trigonometry 2e PDF (Free)

This book is free from its official source, OpenStax. Click below to download the complete PDF — a free web-based reader edition (with per-section, linkable pages) is also available on the OpenStax site if you’d rather read online.

How to Study This Book

If your algebra is already solid, skip straight to Chapter 2 — Chapter 1 (Prerequisites) is a review chapter, and its own key concepts pages make it easy to spot-check any one topic (exponents, factoring, rational expressions) if you get stuck later rather than reading it cover to cover first.

Chapters 2 and 3 are the real foundation. Chapter 2’s equation-and-inequality techniques and Chapter 3’s formal definition of a function are assumed everywhere after this point — do not skip Chapter 3’s sections on domain/range, composition, and transformations even if they feel slow, because Chapters 4 through 6 are, structurally, the same transformation ideas applied to three different function families in a row.

Chapters 4, 5, and 6 (linear, then polynomial/rational, then exponential/logarithmic functions) should be read in that order the first time through: each one reuses the graphing-transformation vocabulary Chapter 3 sets up, and Chapter 6’s logarithm rules assume you are already comfortable manipulating exponents from Chapter 5.

Chapters 7 through 10 form a self-contained trigonometry block. Chapter 7 (the unit circle and right-triangle trig) is the foundation every later trig chapter depends on — Chapter 8’s graphs, Chapter 9’s identities, and Chapter 10’s law-of-sines/cosines and polar/parametric/vector material all reuse Chapter 7’s definitions directly. If your course only needs trigonometry (not the algebra chapters), you can start at Chapter 7 and treat Chapters 1–6 as a reference.

Chapter 9 (identities and equations) is the chapter students find hardest, because it is algebraic manipulation rather than direct computation. Budget real extra time here, and lean on the “convert everything to sine and cosine” strategy in its Memory Tip above before trying anything more advanced.

Chapter 11 (systems and matrices) and Chapter 13 (sequences, probability, counting) are largely independent of the trigonometry block — both can be read any time after Chapter 2, and Chapter 13 in particular pairs naturally with a discrete mathematics course if your program has one.

Chapter 12 (analytic geometry / conics) leans on Chapter 2’s quadratic-equation work and, in its last section, Chapter 10’s polar coordinates — read it after both rather than in strict numeric order if your course sequences it earlier.

Whichever chapter you’re on, work through its own Key Concepts review page (linked at the end of every OpenStax chapter) before attempting the Chapter Review Exercises — it is the book’s own compressed summary of exactly what you need to have retained.

Used In These Programs

This book is used for the Algebra and Trigonometry course in BS Mathematics programs, and as a precalculus-level refresher for BS Computer Science and BS Information Technology students preparing for Calculus. Browse all General Mathematics books or all Mathematics books.

Who Should Read This

This book is written for BS Mathematics students taking a combined Algebra and Trigonometry course, and for BSCS, BSIT, or Engineering students who need a precalculus-level refresher before Calculus I — particularly the function (Chapter 3), exponential/logarithmic (Chapter 6), and trigonometry (Chapters 7–10) chapters, which most calculus courses assume. It is not a from-scratch text for a student with no prior algebra exposure at all: Chapter 1 reviews rather than teaches the basics of exponents, polynomials, and factoring, so a reader who has never taken an algebra course should start with a foundational algebra text and come back to this one for the function and trigonometry material it covers that a purely algebra-focused book does not. Once you have this book’s trigonometry chapters under control, the natural next step for a BS Mathematics student is Calculus Volume 1, which assumes exactly the function, exponential, and trigonometric background this book builds.

Applicable Universities

This book is useful for BS Mathematics students at Pakistani universities offering the degree, including Punjab University, Virtual University, COMSATS, FAST, UET, NUST, and other HEC-recognized institutions, and as precalculus preparation for BSCS/BSIT students at the same universities.

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Related Books

• Calculus Volume 1 – Strang & Herman (OpenStax)
• Calculus Volume 2 – Strang & Herman (OpenStax)
• Calculus Volume 3 – Strang & Herman (OpenStax)
• Discrete Mathematics: An Open Introduction – Oscar Levin (4th Edition)

Algebra and Trigonometry 2e gives BS Mathematics students both halves of their precalculus foundation — algebra and trigonometry — in one consistently structured OpenStax text, ready for a Calculus I course to build on directly. Browse more Mathematics books for the rest of your semester.

Algebra and Trigonometry 2e, by Jay Abramson and contributing authors. OpenStax, Rice University. Free under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 licence. Access for free at https://openstax.org/details/books/algebra-and-trigonometry-2e