TRIGONOMETRY 11TH - 12TH

MARY BORNE

Overview

The students work in groups to graph groups of polar equations and
answer questions concerning these classic curves. After discussing the
answers, the students are then required to develop equations for given
graphs of classic curves.

Expected outcome

The students should be able to recognize the equations of different
classic curves by the position and value of the numbers, cosine and sine.
They should also be able to determine how the graph of a polar equation
will look by the value of these terms.

Background information

This activity should be used in a trigonometry class that has completed
the unit on the trigonometric functions sine and cosine. The students and
teacher should be able to use graphing calculators in polar mode. The students
should also be able to graph a point given the polar coordinates.

Materials

Each student or student group needs a graphing calculator, worksheets,
and a pencil.

The students will also need to refer to their textbook, __Advanced
Mathematical Concepts__ .

Standards

The NTCM Standards encourages a shift towards more active student involvement
in the learning process and the use of technology. This lesson provides
a student centered experience that extends logical reasoning skills by
investigating the connection between the equations of a polar equation
and its graph. Mathematics as Connections encourages students to analyze
and identify important mathematical relationships using a graphing calculator.
The lesson also incorporates Mathematics as Communications by having students
clarify their thinking by formulating ideas in writing.

Engagement

Graphing the first polar equation on the overhead graphing calculator
will show the students what they need to do and how to proceed. I will
then discuss with the class what to look for and how to answer the questions.
The shapes of the classic curves encourages students to continue the activity
to see the different shapes they can form on their calculators.

Exploration

The student will be placed into groups and given the graphing polar
equations worksheet. I will give the students instructions and graph the
first polar equation to show the students how to proceed. The student groups
will work together to discover the classic curves obtained when the polar
equations are graphed. We will discuss the answers to the worksheet together
on the following day. The students will be required to complete a second
worksheet to identify the equations of graphs of classic curves.

Explanation

The first worksheet helps the students to discover how the position
and value of the numbers and trig functions affect the graphs of the classic
curves. The following class is used to determine the equations of the classic
curves given their graphs. The second worksheet is used to help the students
recognize what they have learned.

Evaluation

The second worksheet will be collected and graded. The students will
continue to work in the assigned groups and each student will hand in their
own worksheet for a grade. The skills and knowledge gained in this lesson
will be tested on the polar coordinates chapter test.

Extensions

Graphing the polar equations will help the students to make the connection
when they are learning to change polar coordinates to rectangular coordinates
and back The students can visually see the points on the polar axis and
compare the point on the rectangular axis.

References
__Advanced Mathematical Concepts__ Merrill Publishing
__Trigonometry Activities: For the TI82 & TI85 Graphing Calculators__

by C. R. Dennis & L. M. Neal , International Thomas Publishing

Section ( 9 - 2 ) Name________________________

Graphing polar equations Partner_______________________

Sketch a graph of each equation and complete the following:

1) r = 8 cos x
2) r = 6 sin x

If r = 2a cos x , then a determines___________________________________________________

2a determines______________________________________________________

sin and cos determine _______________________________________________

3) r = 1 + 2 cos x
4) r = 2 + 4 sin x
5) r = 2 + 2 cos x

If r = a + b cos x , then sin and cos determine__________________________________________

a determines_____________________________________________________

a + b determines__________________________________________________

a = b determines__________________________________________________

6) r =
9 sin 2 x

In r = a cos 2 x , a determines ___________________________________________________

the number of petals is determined by_______________________________

7) r = 2 cos 5x 8) r = 3 sin 3x

9) r = 3 cos 4 x
10) r = 5 sin 6x

If r = a cos nx and n is odd , then the number of petals is determined by _____________________

a determines_____________________________________________

If r = a cos nx and n is even, then the number of petals is determined by _____________________

When n is odd there are ____ petals and when n is even
there are _____ petals

Using the classic curves on page 241, name each of the graphs you have sketched:

1)___________________________ 2)___________________________

3)___________________________ 4)___________________________

5)___________________________ 6)___________________________

7)___________________________ 8)___________________________

9)___________________________ 10)___________________________