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Trigonometry

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Trigonometry Cloud County Community College Spring, 2012 Instructor: Timothy L. Warkentin – PowerPoint PPT presentation

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Title: Trigonometry


1
Trigonometry
  • Cloud County Community College
  • Spring, 2012
  • Instructor Timothy L. Warkentin

2
Course Overview
  • The importance of study and completion of
    homework.
  • Resources on the Cloud website syllabus
    (homework assignments), chapter outlines,
    homework solutions, handouts, and class notes.
  • The importance of memorization in the study of
    Trigonometry.
  • Using technology in Calculus I (TI-84
    calculators, Graphing Calculator, Internet
    Resources). Lab Introduction to Wolfram Alpha

3
Chapter 1 Functions and Graphs
  • 1.1 Equations and Inequalities
  • 1.2 A Two-Dimensional Coordinate System and
    Graphs
  • 1.3 Introduction to Functions
  • 1.4 Properties of Graphs
  • 1.5 The Algebra of Functions
  • 1.6 Inverse Functions
  • 1.7 Modeling Data Using Regression

4
Chapter 1 Overview
  • Chapter 1 reviews important material needed to
    begin a study of Trigonometry. A through
    understanding of functions, function inverses and
    the notation used in their descriptions is an
    essential prerequisite to any understanding of
    Trigonometry.

5
1.1 Equations and Inequalities 1
  • The Complex Number System

6
1.1 Equations and Inequalities 2
  • Three ways of writing set solutions
  • Graphing
  • Interval Notation
  • Set Builder Notation
  • The Absolute Value of a Number the distance the
    number is from the origin.
  • The Distance Between Two Numbers the absolute
    value of the difference between the numbers.
  • Any equation that can be put into the form ax b
    0 is a Linear Equation. Example 1

7
1.1 Equations and Inequalities 3
  • Solving Literal Equations Example 2
  • Clear denominators.
  • Complete multiplications.
  • Separate terms with target variable over the
    equality sign from other terms.
  • Factor out the target variable.
  • Divide to isolate the target variable.
  • Solving Quadratic Equations Examples 3 4
  • Taking square roots.
  • Factoring (Zero Product Property).
  • Completing the Square.
  • Quadratic Formula.

8
1.1 Equations and Inequalities 4
  • Solving Inequalities Examples 5 6
  • When an inequality is multiplied or divided by a
    negative number the direction of the inequality
    changes.
  • The Critical Point Method.
  • The Graphical Method.
  • Solving Absolute Value Inequalities Examples 7
    8
  • Using sign switches.
  • Using the distance between two points and a
    number line.

9
1.2 A Two-Dimensional Coordinate System and
Graphs 1
  • The Cartesian Coordinate System.
  • The Distance and Midpoint Formulas. Example 1
  • Graphing by using points. Examples 2-4
  • Graphing using the TI-84 calculator. Examples 2-4
  • Finding x (set y 0) and y (set x 0)
    intercepts. Example 5
  • Finding x intercepts with a TI-84 calculator.
    Example 5
  • The Equation of a Circle Example 6
  • Switching from Standard Form to General Form
    (expand the squares)
  • Switching from General Form to Standard Form
    (double completion of the square) Example 7

10
1.3 Introduction to Functions 1
  • Relations, Functions and 1-to-1 Functions

11
1.3 Introduction to Functions 2
  • Domain pre-images, x-values, independent values,
    inputs.
  • Range images, y-values, dependent values,
    outputs.
  • Function Every pre-images has exactly one image.
  • Functions can be described using function
    notation, ordered pairs, Venn diagrams,
    input/output machine diagrams, and tables.
  • Function notation and dummy variables. Example 1
  • Piecewise Functions (TI-84 calculators). Example
    2
  • Identifying Functions (The Vertical Line Test -
    VLT). Examples 3 6

12
1.3 Introduction to Functions 3
  • Domain Issues Example 4
  • Division by Zero
  • Even Roots of Negative Numbers
  • Physical Constraints
  • Graphing functions using points (TI-84 tables)
    Example 5
  • Graphing functions using the TI-84 calculator and
    Graphing Calculator.
  • Increasing and Decreasing Functions.
  • 1-to-1 functions (The Horizontal Line Test HLT)
  • The Greatest Integer Function (The Floor
    Function, TI-84 int( command) Example 7
  • Using functions to solve applications. Examples
    8-10

13
1.4 Properties of Graphs 1
  • Relation Symmetry Examples 1 2
  • y-axis symmetry -x replacing x yields equivalent
    equation.
  • x-axis symmetry -y replacing y yields equivalent
    equation.
  • Origin symmetry -x replacing x -y replacing y
    yields equivalent equation.
  • Function Parity Example 3
  • Even f-x fx
  • Odd f-x -fx
  • Function Translation Example 4
  • Vertical gx fx k is fx translated
    vertically by k units.
  • Horizontal gx fx-h is fx translated
    horizontally by h units.

14
1.4 Properties of Graphs 2
  • Function Reflection Example 5
  • -fx is fx reflected over the x-axis.
  • f-x is fx reflected over the y-axis.
  • Function Vertical Elasticity Example 6
  • If a gt 1 then gx afx is fx stretched
    vertically by a factor of a.
  • If 0 lt a lt 1 then gx afx is fx
    compressed vertically by a factor of a.

15
1.4 Properties of Graphs 3
  • Function Horizontal Elasticity Example 7
  • If a gt 1 then gx fax is fx compressed
    horizontally by a factor of 1/a.
  • If 0 lt a lt 1 then gx fax is fx stretched
    horizontally by a factor of 1/a.
  • Function Transformation
  • a is the vertical scaling factor.
  • w is the horizontal scaling factor.
  • h is the horizontal shift.
  • k is the vertical shift.

16
1.5 The Algebra of Functions 1
  • Operations on Functions Examples 2, 5, 6 7
  • Domains of Combined Functions Example 1
  • The Difference Quotient Examples 3 4

17
1.6 Inverse Functions 1
  • The inverse of a Relation is that Relation that
    switches the order of the ordered pair elements.
    Every Relation has an Inverse.
  • A Function will have an Inverse Function IFF it
    is a 1-to-1 Function.
  • Identifying 1-to-1 Functions (The Horizontal Line
    Test - HLT). Example 1
  • Proving that a pair of functions are inverses.
    Example 2
  • Finding an Inverse (Switch Method). Examples 3
    4
  • Restricting the domain of a function (domain
    surgery). Examples 5 6

18
1.7 Modeling Data Using Regression 1
  • Linear Regression Models. Example 1
  • The Correlation Coefficient.
  • The Coefficient of Determination.
  • Quadratic Regression Models. Example 2
  • Using the TI-84 to model data.
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