Learning Resources
Outcomes
In this lesson you will learn
- the relationship between the length of an arc on a circle and the radius of the circle
- a review of radian measure
- the definition of the terminal arm of an angle
By the end of this section students will be able to:
- describe and apply the connection between arc length and radian measure
- demonstrate an understanding for the use of, and need for, radian measure in the domain of trigonometric functions
- model situations with periodic curves
- analyze and solve polynomial, rational, absolute value, and trigonometric equations
- analyze relations, functions and their graphs
- analyze the effect of parameter changes on the graphs of functions and express the changes using transformations
- demonstrate an understanding for asymptotic behaviour
Introduction
You have encountered radian measure in Mathematical Modeling, Book 2. This section of study reviews and reinforces the relationship between radian measure and arc length. You will examine the relationship between the radius of a circle and its circumference to investigate exactly how many radians there are in a circle. The radian measure will be used to calculate arc length and it will also be used as the domain of periodic functions.
One of the core trigonometric topics you have previously studied is that of periodic functions; those that recur in cycles. Think for a moment about your school timetable, the phases of the moon and the ocean tides. These are all examples of periodic phenomena since they occur in repeated and expected cycles.
The sine and cosine functions will be revisited and their symmetry investigated. Hopefully, this will serve to reinforce the connection between the sine and cosine functions. Then the tangent function will be investigated as a combination of the sine and cosine functions through the solving of a problem involving a Ferris wheel. Sound familiar? By now, your mind should be wandering back to some of the work studied in Mathematics 2205.
By the end of this section, you should have a solid understanding of all three primary trigonometric functions.
The following materials are required for the first Investigation. Be sure to have them on hand.
- graph paper
- compass
- protractor
This unit of study requires 5 to 6 hours for completion.
Prerequisites
To be successful in this lesson, it would be helpful to know the following:
- formula for circumference of a circle
- the meaning of concentric circles
- coordinates on the unit circle in terms of cosine and sine
- periodic functions
- fractional computations
- the sine and cosine graphs
- radian measure