Mathematics

Syllabus

Mathematics 37 – Trigonometry with Geometry

Effective Fall 2007

USING SCHAUM’S OUTLINE FOR THE GEOMETRY PORTION

Texts: 1)         Schaum’s Outlines Geometry,  Third Edition; Barnett Rich; McGraw-Hill

            2)         Trigonometry, Second Edition; Dugopolski; Pearson Addison Wesley

Supplementary Materials for the Student (for trigonometry text only):

Student Solutions Manual — Odd problem solutions

Comments:

           If you have not taught Geometry before, you may wish to use the alternate syllabus that uses College Geometry as the text for the geometry part of the course.

            I have supplemented the schedule with some comments on how I have often handled things teaching Math 37.  Use what you wish, and change or ignore the rest.  If I say you need a certain theorem, it means that in class I did something between a quick oral mention of it with a brief sketchy or intuitive discussion of proof and writing it down and doing a complete proof.

            Be aware that the Outline groups definitions, postulates, and theorems under the name “principle.”  You will need to distinguish among these for your students.  Sometimes you will need to reorder the principles so as to define, assume, and prove in a logical order. 

---  Joe Berland

SCHAUMS:

Total hours: 4

3.2/3.3:  I covered the following (some is not in text at this point):

            Undefined terms:           Point, line, plane, between

            Postulates:        1)         Through any two points, there is one and only one line.

                                    2)         Through any three noncollinear points, there is one and only

                                                one plane.

            Theorem:                      Two lines intersect in at most one point.

            Definitions:        Line segment, notation: AB means length of , congruent

                        segments: , ray, bisector of a segment,

                        trisector of a segment

3.5:      I skipped “reflex angle.”

3.7:      In 3.7B, I did only principles 2 (to be proven later) and 6.

14.3:    You will need to write HW problems that don’t depend on geometry not learned

yet (they don’t have to be about geometry at all).

Total hours: 3

4.2:      I did only postulates 1-7.  “Halves of equals are equal” is a convenient corollary

to postulate 7 to give students.  Postulates 16-19 can be saved until needed to prove something later.  The following theorems (not in text) are important:

1)         Segment Addition Theorem:  If  P  is between  A  and  B, Q  is between  C  and  D, .

2)         Segment Subtraction Theorem:  Similar

3)         Angle Addition Theorem:  If  P  is on the interior of , Q  is on the interior of , , then .

4)         Angle Subtraction Theorem:  Similar

4.3:      I skipped principle 2.  Useful theorems to add to text:

                                    1)         Sets of perpendicular lines form congruent angles.

                                    2)         If two lines form a pair of congruent adjacent angles, then the lines are

                                                perpendicular.

Total hours: 4

5.2:      To prove principle 1, you will need this theorem:  An angle has one an only one

bisector.

Total hours: 5

Indirect Proofs: You’ll need to make up some problems. See me if you want my set.

6.1:      An indirect proof of principle 5 requires this theorem:  Through a point off a line, there is one and only one perpendicular to the line.  Principle 5 can be used to prove principle 3, from which principles 2 and 4 follow.  Principle 8 can be proven using principle 3 to lead to a contradiction of the Parallel Lines Postulate.  From principle 8, principles 7 and 9 follow.  I omitted principles 10, 11, and 12.

6.3:      I did principles 1, 4, and 6 only.

6.4:      Give definition for a convex polygon.  Principles 1 and 2 should be corrected to refer to convex polygons only.

Total hours: 2

8.1:      I added the following definitions here:

                                    1)         Measure of a minor arc:  The measure of its central angle

                                    2)         Measure of major arc  :   (measure of minor arc )

                        I did principles 4, 6, 7, 8, 9, and 10 only.

8.2:      I did principles 1,2, and 4 only.

8.3:      I did principle 1 during 8.1.  Here I did principles 2, 4, 6, 7, 8, and 9 only.  If you

assign HW using a trapezoid, you’ll need to define it.

Total hours: 4

9.4:      I did principle 3 only, but you might also do principles 4 and 5.  You can do principle 9, or prove it later as part of proving Pythagorean Theorem.  You need this theorem:  If .  If you assign HW using parallelogram or trapezoid, you will need to define.

9.3:      Need theorem:  If  then the following are equivalent:

 .  Then I did principle 1 only.

9.9:      I proved principle 2 from 7.8 as part of proving Pythagorean Theorem; you could

do it separately.  You need converse of Pythagorean Theorem:  If  a, b, and  c  are the lengths of three sides of a triangle, and , then the triangle is a right triangle.

DUGOPOLSKI:

Chapter 1: Angles & the Trig Functions

Total hours: 9

1.2:  Area of a sector problems (like Exercises 91-94, 101-104) are done later in the course when you return to geometry. You can define “circumference” loosely as “distance around” (more precise definition later in course). Need theorem (need not prove): There exists a constant  such that if  C  and  d  are respectively the circumference and diameter of any given circle, then . Need theorem:  .  Need theorem:   (to be argued later in course if time).

Chapter 2: Graphing Trig Functions

Total hours: 8

Chapter 3: Trig Identities

Total hours: 8  

Chapter 4: Inverse Trig Functions; Solving Conditional Trig Equations

Total hours: 6

Chapters 5 & 6: Law of Sines & Cosines; Polar Coordinates

Total hours: 6

5.3:  The types of problems in this section are covered later in the course when you return to geometry, so this section may be skipped now.

SCHAUM’S:

Total hours: 4

2.16:    To prove distance formula, you need to define rectangle and prove that opposite sides of a rectangle are congruent.

2.17:    You can take principle 1 as the definition of slope, from which principle 3 follows as a theorem. Principles 8-11 follow from principle 3 (for 10 and 11, you can first show that for one angle  between ).  I skipped principle 13.

Analytic Geometry/Ch.7:             I defined and showed how to label coordinates for quadrilateral, rectangle, parallelogram, trapezoid, isosceles trapezoid, and triangle.  Use definitions only to justify coordinates to avoid circular reasoning in coordinate proofs.  For HW, we proved eight theorems from Ch.7 by coordinate geometry (see attached assignment).  I defined rhombus, proved synthetically it’s a parallelogram and that its diagonals are perpendicular (you could do a coordinate proof).  I synthetically proved 7.2 principle 4.

Total hours: 2

10.3:    Make some HW to find areas of parallelograms and triangles given two sides and an included angle, and of triangles given the lengths of the three sides (using trig, not Heron’s Formula).

Total hours: 3

11.1:    First prove principle 2.  Then you can define center of regular polygon as center of circumscribed circle.  I skipped principles 3, 4, 5.  Principle 10 done previously.  Some principles I saved until needed in upcoming proofs.

11.5:    You need the following theorems:

1)         If a circle is divided into n congruent arcs, the segments joining the

endpoints of the arcs form a regular polygon of  n sides (and all such polygons

are congruent), and tangents drawn at the endpoints of the arcs form a regular polygon of  n sides (and all such polygons are congruent).

2)         (Provable in Calculus):  For a given circle, let  be the perimeter of any

inscribed regular polygon of  n sides.  There exists a unique number  C  (Def:  circumference of the circle) such that as n grows arbitrarily large,  becomes arbitrarily close to  C.  We say .                 

3)         (Provable in Calculus):  For a given circle, let  be the perimeter of any

circumscribed   regular polygon of  n sides.  Then .

4)         (Argue intuitively):  For any integer n, , no matter how large,

            .

            Then restate  theorem (I don’t attempt to prove this) and restate theorem.  Then you also need the following the

1)         (Provable in Calculus):  For a given circle, let  be the area of any inscribed

regular polygon of  n sides.  There exists a unique number A (Def:  Area of the circle) such that .

2)         (Argue as the author does on P. 180):  If  A and r are respectively the area and

radius of any given circle, then .

3)         (Argue like previous theorem except using circumscribed polygons):  For a

given circle, let  be the area of any circumscribed regular polygon of  n sides.  Then .

4)         (Argue intuitively):  For any integer n, , no matter how large,

.

11.6:    I did principles 1, 2, and 4 only. 

11.7:    I usually skip this section because of time.  Good to do if you have time.

Approximation of : 

For a polygon of n sides circumscribed about a circle, let  be the angle formed by an apothem and an adjacent radius of the polygon.  Then, if r is the radius of the circle, .  Similarly, for an inscribed polygon, .  Thus, .  With calculators, if students let , etc., they get a better and better decimal approximation of .

Total hours: 2  

The chapter on volumes and surface areas has been omitted from this edition. You will need to make your own HW problems (my HW assignment is attached.) I taught the following surface areas:

                        1)         Lateral area of right prism:   (perimeter x height).  This leads to:

                                    Lateral area of right cylinder:  .

            2)         Lateral area of frustum of regular pyramid: 

                                    = (sum of perimeters of bases)(slant height).  This leads to: 

                                    Lateral area of frustum of right cone:  .  This

leads to:  Lateral area of right cone:  .

            3)         Area of sphere:  (I don’t try to justify this.)

I did volumes in this order:

                        1)         Prism (which includes rectangular solid which includes cube) leads to

cylinder.

                        2)         Cone:  If you have time, you can estimate the volume of the cone formed

by revolving the triangle with vertices (0,0), (8,0), (0,8) about the  y-axis, by summing the volumes of four cylinders (I put radii at midpoints of subintervals).  Then find volume of cylinder, r = 8, h = 8.  Note that

.  We suspect in general, and Calculus proves, that

V(cone) = V(cylinder) = . 

3)         Sphere:  Egl came up with a great method for finding the volume of a sphere by using the volumes of a cylinder and a cone. See me if you want and I’ll show it to you.

TOTAL HOURS:  70

Possible exam schedule:

Exam 1:            Schaum’s:  14.2 through Indirect Proofs (following the order of the syllabus)

Exam 2:            Schaum’s:  6.1 through 9.10 (following the order of the syllabus)

Exam 3:            Dugopolski:  1.1 – 1.6

Exam 4:            Dugopolski:  2.1 – 2.4, 3.1 – 3.5

Exam 5:            Dugopolski:  4.1 – 4.4, 5.1, 5.2, 6.4

Quiz:                Schaum’s covered after Exam 5

Final:                Comprehensive for the trigonometry, with a little geometry to keep them

honest.

Joe Berland, May 2007