Module 5-II

 Trigonometric Functions

 

Introduction

 

If we want to use the trigonometric ratios to define functions of a real number variable, then it is necessary to decide what system of units is to be used to relate angles and numbers.  Because of the way it relates angles and distances, the radian measure of angles turns out to be more convenient for many applications (especially in Calculus), so, in Section 5.3 of the text, the trigonometric functions of a real variable, t, are defined as the corresponding trig. ratios for an angle of t radians.

            i.e.     , etc.

 

Another way of stating the above definition is to say that cos(t) and sin(t) are the x and y coordinates of a point on the unit circle at a distance of t units from the positive x axis as measured counterclockwise around the circumference of the circle.

This “unit circle definition” has the advantage that it extends very naturally to non acute angles and indeed allows us to define cos(t) and sin(t) for arbitrary real values of t.

It also makes sense of a lot of rules and techniques that otherwise would seem arbitrary and hard to remember

 

 

Learning Objectives

 

In this Module you will study the trigonometric functions of a real (number) variable.

After completing this Module you will be able to:

 

·  State the “unit circle” definitions of the trigonometric functions

·  Establish whether given trigonometric equations are or are not identities

·  Solve conditional equations and inequalities involving trig functions

·  Sketch the graphs of trigonometric functions and of others related to them by simple operations

·  Determine possible equations for given graphs

·  Find expressions in terms of trigonometric functions to model various kinds of cyclical phenomena, and use the above skills to solve applied problems involving such phenomena

 

Module Outline and Section Learning Objectives

 

            Section 1

 Definitions and Evaluation

·        State the “unit circle” definitions of the trigonometric functions

·        Use the definitions to evaluate these functions (exactly if possible)

 

            Section 2

 Graphs of Trigonometric Functions

·        Sketch graphs of trig functions and of others related to them by composition with linear functions

·        Identify equations from given graphs and from verbal descriptions of periodic phenomena

 

            Section 3

 Trigonometric Identities

·        State and use the pythagorean identity, periodicity and symmetry properties, complementarity relations, and sum rules

·        Prove these and other identities and/or identify counterexamples for equations that are not identities

 

            Section 4

 Inverse Trig Functions, Equations and Inequalities

·        Solve simple trig equations either by recognizing particular special cases and/or by reference to “inverse trig functions”

·        Use identities and algebraic manipulation to solve more complicated equations

·        Solve various applied problems involving trigonometric equations and inequalities