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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.

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.