# MAT1093

## Precalculus

1093. Precalculus. (3-0) 3 Credit Hours. (TCCN = MATH 2312)

Prerequisite: MAT 1073 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: DL01 \$75; LRC1 \$12; LRS1 \$45; STSI \$21.

## Lesson Plan

Date Sections Topics Prerequisite Skills Student learning outcomes
Week 1 Orientation
• Introduction to MyMathLab
Week 1 1.3 Functions and their graphs
• 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 Trigonometric 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 Trigonometric 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
• 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 Graph sinusoidal functions of the form y = A sin (ωx – φ) + B and y = A cos (ωx – φ) using transformations and determine the amplitude, \abs(A), period, T, and phase shift, φ/ω
Week 5 Test 1 Review Session. Common Test 1: Ch.1 and 2.
Week 6 3.1 The inverse Sine, Cosine and Tangent functions
• 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 Secant, Cosecant and Cotangent 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 Section A.4: Solving algebraic equations Find exact solutions in the interval [0, 2π) and in general form for equations with single trig function
Week 7 3.3B Trigonometric Equations
• 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 Trigonometric Identities
Prove simple identities using the fundamental identities and algebraic technics
Week 8 3.5 Sum and Difference Formulas Section 2.2: Trigonometric Functions: Unit Circle Approach
• 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
• 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
• 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
• Test 2 Review Session
• Common Test 2: Chapter 3
Week 10 4.1 Right triangle definitions of trig functions and related applications
• 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 10 4.2 The Law of Sines
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: Trigonometric 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
• 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
• 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: Trigonometric Functions: Unit Circle Approach 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 7.4 Logarithmic and Exponential Equations
• Section A-1: Law of Exponents
• Section 7.1: Exponential functions
• Section 7.2: Logarithmic functions
• Section 7.3: Properties of logarithms (Review in class as needed)
Find exact and approximate solution sets for exponential and logarithmic equations of any base, including those from application questions
Week 14 7.6 Exponential growth and decay models
• Section A-4: Solving quadratic equations
• Section 7.1: Exponential functions
• Section 7.2: Logarithmic functions
• Section 7.3: Properties of logarithms
Create and use exponential growth and decay models from two data points
Week 14 7.6 Newton’s law of Cooling models 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 models 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