MAT1093

Precalculus

(3-0) 3 Credit Hours. (TCCN = MATH 2312)

Prerequisite: MAT1073 or the equivalent course or satisfactory performance on a placement examination. Exponential functions, logarithmic functions, trigonometric functions, complex numbers, DeMoivre’s theorem, and polar coordinates. May apply toward the Core Curriculum requirement in Mathematics. Generally offered: Fall, Spring, Summer. Course Fees: LRC1 $12; LRS1 $15; STSI $15.

Date	Sections	Topics	Prerequisite Skills	Student learning outcomes
Week 1	Orientation	Distribute and read syllabus Introduction to MyMathLab
Week 1	1.3	Functions and their graphs	Interval notation Solving linear equations and inequalities Evaluating algebraic expressions	Determine whether a relation is a function Find the Difference Quotient of a simple quadratic or radical function Find the domain of a function defined by an equation or a graph Identify the graph of a function and get information from the graph
Week 2	1.7	One-to-one functions	Section 1.3: Functions and their graphs	Determine when a function or its graph is one-to-one
Week 2	1.7	Inverse functions	Section 1.3: Functions and their graphs	Find the inverse of a function defined by a graph or an equation Use the composition property to verify two functions are the inverses of each other Find the inverse of a function algebraically or graphically
Week 2	2.1	Angles and their measure	Elementary geometry and terminology	Know the definition of an angle in standard position and when its measure is positive or negative Know relationship between degrees and radians and be able to sketch angles of any measure Be able to convert angles to and from decimal degrees and D-M-S notations Know formulas for finding the length of a circular arc and the area of a sector of a circle Find the distance between two cities at same longitudes and at different longitudes Know the formula relating linear speed of an object in circular motion with its angular velocity in either radians per unit of time or revolutions per unit of time or vice versa
Week 3	2.2	Trig. Functions: Unit Circle Approach	Appendix A.2: Geometry Essentials Section 1.2: Symmetry of graphs	Learn the definitions of the six trig functions as derived from the Unit Circle and apply them to find exact values for a given point on this circle Use the Unit Circle definitions to find the exact values for all six trig functions for angles of π/4, π/6 and π/3 radians, and integer multiples of these angles Use a course-approved scientific calculator to approximate values for the six trig functions of any angle Learn the definitions of the six trig. functions derived from a circle of any radius r, and use them to find exact and approximate values of these functions for a given point on the circle, including those in application questions
Week 3	2.3	Properties of the Trig. Functions	Section 1.3: Functions and their graphs Section 1.4: Even and Odd Functions	Determine the domain and range of each of the six trig functions, their period, and their signs in a given quadrant of the x-y plane Learn the reciprocal and quotient identities based on the definitions from the Unit Circle of the six trigonometric functions Use the Unit Circle to derive the three Pythagorean Identities to complete the set of Fundamental Identities Find the exact value of the remaining trig functions, given the value of one and the sign of another, using either a circle of radius r or the Fundamental Identities Determine and use the Even-Odd properties to find exact values for the six trigonometric functions
Week 4	2.4	Graphs of the Sine and Cosine Functions	Algebraic graphing technics and transformations	Graph on the x-y plane the functions f(x) = sin x and f(x) = cos x using key points Graph functions of the form y = A sin (ωx) and y = A cos (ωx) using transformations Determine the Amplitude and Period of sinusoidal functions from equations and graphs Find equations of sinusoidal functions given their graphs
Week 4	2.5	Graphs of the Tangent, Cotangent, Cosecant and Secant Functions	Finding Vertical asymptotes of rational functions	Graph the basic tangent, cotangent, secant and cosecant functions using key points, vertical asymptotes, and reciprocal identities, as needed
Week 5	2.6	Phase shift and Applications	Algebraic graphing technics and transformations	A|, period, T, and phase shift, φ/ω
Week 6	3.1	The inverse sine, cosine and tangent functions	Section 1.7: Inverse functions Section 2.2: Values of trig functionsValues of trig functions Section 2.3: Properties of the Trig. Functions Section 2.4: Graphs of the Sine and Cosine Functions Section 2.5: Graph of the Tangent Function Solving algebraic equations	Determine the inverse functions for the sine, cosine and tangent knowing their restricted domains that make these functions one-to-one Find the exact values of a given inverse sine, cosine or tangent function knowing that each inverse function represents an angle Use approved scientific calculator to estimate sine, cosine and tangent functions Use properties of inverse functions to find exact values for certain composite functions For a given sine, cosine or tangent function find the inverse function algebraically and its domain Solve simple equations that contain inverse trigonometric functions, including some from applications
Week 6	3.2	The inverse trig functions continued (Secant, Cosecant and Cotangent)	Section 1.7, Inverse functions Section 2.3: Properties of the Trig. Functions Section 2.5: Graphs of the Cotangent, Cosecant and Secant Functions	Find the exact value of composite expressions involving the inverse sine, cosine or tangent function Know definitions for the inverse secant, cosecant and cotangent functions, including their domain and range, and determine their exact and approximate values Write composite functions of trigonometric and inverse trigonometric functions as an Algebraic expression
Week 6	3.3A	Trigonometric equations involving a single trig function	Week	* Find exact solutions in the interval [0, 2π) and in general form for equations with single trig function
Week 7	3.3B	Trig Equations	Section A.4: Solving algebraic equations Section 2.2: Trig. Functions: Values of trig functions	Solve linear, quadratic and other equations containing trigonometric functions, including those from application questions and those that can be solved using the Fundamental Identities Find exact solutions in the interval [0, 2π) and in general form Use a course-approved calculator to find approximate solutions of trigonometric equations that require the use of an inverse function
Week 7	3.4	Trig. Identities	* Section 2.3: Fundamental Identities and even-odd properties Fundamental Identities and even-odd properties Algebraic operations with fractions, polynomials and factoring polynomials	Prove simple identities using the fundamental identities and algebraic technics
Week 8	3.5	Sum and Difference Formulas	Section 2.2: Trig. Functions: Values of Trig functions	Use sum and difference formulas to find exact values, establish identities and evaluate compositions with inverse functions Solve trigonometric equations linear in both sine and cosine
Week 8	3.6A	Double-angle formulas	Section 2.1: Angles and their measure Section 2.3: Finding exact values given the value of a trig function and the quadrant of the angle	Use double-angle formulas to find exact values Use double-angle formulas to solve trigonometric equations (including from applications) Establish identities
Week 8	3.6B	Half-angle formulas	Section 2.1: Angles and their measure Section 2.3: Finding exact values given the value of a trig function and the quadrant of the angle	Use half-angle formulas to find exact values Establish identities
Week 9	3.7	Product-to-Sum and Sum-to-Product Formulas	Basic algebra and geometry	Use product-to-sum and sum-to-product formulas
Week 9/10	4.1	Right triangle definitions of trig functions and related applications	Basic algebra and geometry Section A.2: Pythagorean Theorem Section 3.3: Trig Equations	Learn the definitions of the six trigonometric functions defined using a right triangle and apply them to solve any right triangle given or sketched from application questions Learn how to use bearings in application questions that generate right triangles to be solved using the right triangle definitions of the trig functions
Week 9		Test 2 Review Session Common Test 2: Chapter 3	Week	Week
Week 10	4.2	The Law of Sines	Basic algebra and geometry Section 3.3: Trig Equations || Learn and use the Law of Sines to solve two cases of oblique triangles (ASA and SAA for case 1, and SAA for case 2, also known as the ambiguous case that can result in no solution, one solution or two solutions) and related applications questions including those with bearings
Week 11	4.3	The Law of Cosines	Section 3.3: Trig Equations	Use the Law of Cosines to solve the other two cases of oblique triangles (SAS for case 3 and SSS for case 4) and related applications questions including those with bearings
Week 11	4.4	Area of a Triangle	Section A.2: Geometry Essentials	Find the area of a SAS triangle using the sine function to find the altitude Find the area of a SSS triangle using Heron’s Formula
Week 11	5.1	Polar Coordinates	* Section 1.1: Rectangular coordinates Section 2.2: Definition of the trig functions Section 3.1: Inverse Functions	Plot points using polar coordinates and find several polar coordinates of a single point Convert polar coordinates to rectangular coordinates and vice versa Transform equations from polar form to rectangular form and vice versa
Week 11	5.2	Polar Equations and Graphs	Section A-3: Completing the square Section 1.2: Graphing lines and circles	Graph simple polar equations by converting them to rectangular form and then use Algebra to graph this rectangular equations
Week 11/12	5.3	The complex plane	Section A.5: Complex numbers Section 2.2: Values of sine and cosine functions	Plot points in the complex plane Convert complex numbers from rectangular to polar/trigonometric form and vice-versa
Week 12	5.3	DeMoivere’s Theorem	Section 2.2: Values of sine and cosine functions	Use the trigonometric form of complex numbers to multiply, divide, and raise them to natural powers
Week 12	Test 3 Review Session Common Test 3: Ch.4 and 5
Week 13	7.1	Exponential functions	Exponents Section 1.6: Graphing technics and transformation Section A-4: Solving equations	Evaluate exponential expressions, including those with the natural base, e, using an approved scientific calculator Graph a simple exponential equation and observe its domain, range, y intercept, horizontal asymptote, and that the graph is a smooth and continuous curve that is increasing everywhere Solve simple exponential equations by equating the exponents of two equal exponential expressions of the same base
Week 13	7.2	Logarithmic functions	Exponents Section 1.6: Graphing technics and transformation Section A-4: Solving equations	Note that logarithmic functions are inverse functions of exponential functions and change exponential expressions to equivalent logarithmic expressions and viceversa Graph logarithmic functions and observe their domain and range Evaluate common and natural logarithms using an approved scientific calculator Solve base 10 and base e single log equations by changing them to equivalent exponential form and checking for extraneous solutions Determine the domain of any logarithmic function
Week 13	7.3	Properties of logarithms	Exponents Section 1.6: Graphing technics and transformation Section A-4: Solving equations	Use properties of logarithms to write a logarithmic expression as a sum or difference of simple logarithms and vice-versa Use the change of base formula to evaluate logarithms whose base is not ten or the natural number, e
Week 13	7.4	Log and exp equations	* Exponents Section 1.6 Graphing technics and transformation Section A-4 Solving equations || Find exact and approximate solution sets for exponential and logarithmic equations of any base, including those from application questions
Week 14	7.6	Exp. growth and decay models	Section A-4: Solving quadratic equations	Create and use exponential growth and decay models from two data points
Week 14	7.6	Newton’s law of Cooling	Section A-4: Solving quadratic equations	Create and use exponential models based on Newton’s Law of Cooling
Week 14	7.6	Logistic growth and decay	Section A-4: Solving quadratic equations	Use Logistic growth and decay models to find present and future values, and times for any future value
Week 15		Common Final Exam Review	All topics covered during the semester