college algebra and trigonometry, ratti and … · mat 1570 – trigonometry classes meeting three...

SINCLAIR COMMUNITY COLLEGE DAYTON, OHIO

DEPARTMENT SYLLABUS FOR COURSE IN

MAT 1570 - TRIGONOMETRY

(3 SEMESTER HOURS)

1. COURSE DESCRIPTION: Trigonometric functions of angles, solving right and oblique triangles,

identities, trigonometric and inverse trigonometric equations, vectors,

radian measure, graphs of trigonometric functions and inverse

trigonometric functions.

2. COURSE OBJECTIVES: To increase the level of mathematical maturity of the student, to expose

the student to further applications of mathematics, and to prepare the

student for the study of calculus and courses in other disciplines that

require the use of trigonometry.

3. PREREQUISITE: Grade of “C” or better in Mat 1470 or satisfactory score on the

Mathematics Placement Test.

4. ASSESSMENT: In addition to required exams as specified on the syllabus, instructors are

encouraged to include other components in computing final course

grades such as homework, quizzes and/or other special projects.

However, 80% of student’s course grade must be based on in-class

proctored exams.

5. TEXT: COLLEGE ALGEBRA AND TRIGONOMETRY, 3th Edition

Ratti and McWaters

Pearson

Adopted: Spring 2016

MyMathLab is a required component of this course. It will give

students access to the online version of the textbook, as well as a

required set of homework assignments and quizzes.

6. CALCULATOR POLICY: A scientific calculator is required. Graphing calculators are not allowed

on exams.

7. INTERNSHIPS Please include the following in your syllabus:

Experiencing an internship in your field of study is the best way to begin

a career. Companies offer opportunities throughout the year for students

to practice what they learned in the classroom to solve real world of

work problems. To learn more about internship opportunities and how

to connect your skills with a future employer, contact Chad R.

Bridgman, M.S.M. Internship Coordinator for Science, Mathematics &

Engineering by phone 937-512-2508, office (3-134), or email

[email protected], and begin test driving your future career!

8. PREPARED BY: Kinga Oliver – point of contact, Najat Baji, Craig Birkemeier, Kay

Cornelius, David Hare, Susan Harris, David Stott

Effective: Spring 2016

mailto:[email protected]


COURSE SCHEDULE FOR COURSE IN

MAT 1570 – TRIGONOMETRY

(3 SEMESTER HOURS)

CLASSES MEETING THREE TIMES PER WEEK

Lecture Sections Topics

1

5.1

Intro

Angles and Their Measure

2 5.1 Angles and Their Measure

3 5.2 Right-Triangle Trigonometry


5 5.3 Trigonometric Functions of Any Angle; The Unit Circle


7 Holiday/Catch-up


9 5.4 Graphs of Sine and Cosine Functions


11 5.5 Graphs of the Other Trigonometric Functions


13 5.6 Inverse Trigonometric Functions


15 Catch-up/Review

16 Review

17 Test #1: Chapter 5

18 6.1 Verifying Identities


20 6.2 Sum and Difference Formulas

21 6.2

6.3

Sum and Difference Formulas

Double-Angle and Half-Angle Formulas

22 6.3 Double-Angle and Half-Angle Formulas

23 6.4 Product-to-Sum and Sum-to-Product Formulas

24 6.5 Trigonometric Equations I


CLASSES MEETING THREE TIMES PER WEEK


25 6.5

6.6

Trigonometric Equations I

Trigonometric Equations II

26 6.6 Trigonometric Equations II

27 Catch-up/Review

28 Review


30 7.1 The Law of Sines


32 7.2 The Law of Cosines


34 7.3 Areas of Polygons Using Trigonometry

35 7.4 Vectors

36 7.4 Vectors

37 7.5 The Dot Product


39 7.6 Polar Coordinates



42 7.7 Polar Form of Complex Numbers (excluding DeMoivre’s Theorem)

43 Catch-up/Review

44 Review


46 Review

47 Review

48 Final Exam




(3 SEMESTER HOURS)

CLASSES MEETING TWO TIMES PER WEEK


1

5.1

Intro


2 5.1

5.2


Right-Triangle Trigonometry

3 5.2

5.3


Trigonometric Functions of Any Angle; The Unit Circle


5 5.3

5.4


Graphs of Sine and Cosine Functions



8 5.5

5.6

Graphs of the Other Trigonometric Functions

Inverse Trigonometric Functions


10 Catch-up/Review



13 6.1

6.2

Verifying Identities


14 6.2

6.3



15 6.3

6.4


Product-to-Sum and Sum-to-Product Formulas

16 6.5 Trigonometric Equations I


CLASSES MEETING TWO TIMES PER WEEK



18 Catch-up/Review



21 7.1

7.2

The Law of Sines

The Law of Cosines

22 7.2

7.3

The Law of Cosines

Areas of Polygons Using Trigonometry

23 7.3

7.4


Vectors

24 7.4

7.5

Vectors

The Dot Product





29 Catch-up/Review


31 Review

32 Final Exam

SINCLAIR COMMUNITY COLLEGE

DAYTON, OHIO



(3 SEMESTER HOURS)

SUMMER CLASSES MEETING 2 TIMES PER WEEK


1

5.1

Intro




4 5.3

5.4



5 5.4

5.5



6 5.5

5.6



7 5.6


Catch-up/Review

8

Review

Test #1: Chapter 5


10 6.2

6.3



11 6.3

6.4



12 6.5

6.6



13 6.6


Catch-up/Review

14

Review

Test #2: Chapter 6



Week Sections Topics



17 7.3

7.4


Vectors

18 7.4

7.5

Vectors

The Dot Product

19 7.5

7.6

The Dot Product

Polar Coordinates


21 7.7

Polar Form of Complex Numbers (excluding DeMoivre’s Theorem)

Catch-up/Review

22

Review

Test #3: Chapter 7

23 Review

24 Final Exam

SINCLAIR COMMUNITY COLLEGE

DAYTON, OHIO



(3 SEMESTER HOURS)



1

5.1

Intro


2 5.1

5.2






6 5.3

5.4



7 5.4

5.5




9 5.5

5.6




11 Review



14 6.1

6.2

Verifying Identities


15 6.2

6.3



16 6.3

6.4



17 6.4

6.5



18 6.5

6.6





Week Sections Topics


20 Review



23 7.1

7.2

The Law of Sines

The Law of Cosines

24 Holiday/Catch-up

25 7.2

7.3

The Law of Cosines


26 7.3

7.4


Vectors

27 7.4 Vectors


29 7.5

7.6

The Dot Product

Polar Coordinates


31 7.6

7.7

Polar Coordinates

Polar Form of Complex Numbers (excluding DeMoivre’s Theorem)


33 Review


35 Review

36 Final Exam

MAT 1570 Formulas

Prerequisite Formulas

Formulas of special importance that students are expected to know upon entering this

course.

- Pythagorean Theorem

The square of the length of the hypotenuse of a right triangle equals the sum

of the squares of the lengths of the other two sides of the triangle: 222 bac

- Distance Formula 2

12

2

12 )()( yyxxd

- Sum of Angles

The sum of the measures of the three angles in any triangle is o180 :

oCBA 180

- Similar Triangles

Ratios of the lengths of corresponding sides of similar triangles are equal.

- Quadratic Formula If 02 cbxax , then a

acbbx

2

42 (𝑎 ≠ 0)

- Geometric Formulas Area Perimeter

Triangle bhA2

1 cbaP

Circle A 2r 2C r

- Slope of a Line 12

12

xx

yym

- Definition of i and 2i 1i , 12 i

ANGLE RELATIONSHIPS AND SIMILAR TRIANGLES

Vertical Angles: Vertical angles are opposite each other when two lines cross and have equal measures

(AED=BEC and AEB=DEC)

Alternate Interior Angles For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate interior angles. Alternate interior angles have the same angle measurement. Angle B and angle C are also alternate interior angles.

Alternate Exterior Angles For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle D are called alternate exterior angles. Alternate exterior angles have the same angle measurement. Angle B and angle C are also alternate exterior angles.

Corresponding Angles For any pair of parallel lines 1 and 2, that are both intersected by a third line, such as line 3 in the diagram below, angle A and angle C are called corresponding angles. Corresponding angles have the same angle measurement. Angle B and angle D are also corresponding angles.

Types of Triangles:

a. Equilateral triangle: If the lengths of all three sides of the triangle are equal, then it is called an equilateral triangle. Since the sum of all the angles of a triangle is 1800, it can be said that each angle of an equilateral triangle is 600.

b. Isosceles triangle: If only two sides of a triangle are equal in length, it is called as an isosceles triangle.

c. Scalene triangle : If all the sides of a triangle have different lengths it is called a scalene triangle

d. Acute triangle: A triangle in which all the angles are acute, (i.e. < 900) is called as an acute triangle.

e. Obtuse triangle: A triangle in which one of the angles is obtuse is called as an obtuse triangle.

f. Right Triangle: It is a triangle in which one of the angles is a right angle.

Congruent Triangles - Two triangles are said to be congruent, if all the corresponding parts are equal. The

symbol used for denoting congruence is and PQR STU implies that corresponding angles and corresponding sides are equal.

Similar Triangles - Two triangles are called similar if all their angles are equal,

respectively. Note that it is sufficient for two triangles to have two pairs of equal

angles to be similar. Corresponding sides must be proportional.

Course Formulas- Formulas that students are required to learn in this course.

Chapter 5

1° = 60′, 1′ = 60′′

180° = 𝜋 radians

Arc Length 𝑠 = 𝑟𝜃

Area of a Sector 𝐴 =1

2𝑟2𝜃

Linear Velocity 𝑣 = 𝑠/𝑡

Angular Velocity 𝜔 = 𝜃/𝑡

𝑣 = 𝑟𝜔

Right-Triangle Definitions

sin 𝜃 =𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=

𝑎

𝑐 cos 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒=

𝑏

𝑐 tan 𝜃 =

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡=

𝑎

𝑏

csc 𝜃 =ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒=

𝑐

𝑎 sec 𝜃 =

ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡=

𝑐

𝑏 cot 𝜃 =

𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡

𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒=

𝑏

𝑎

Reciprocal Identities

csc 𝜃 =1

sin 𝜃 sec 𝜃 =

1

cos 𝜃 cot 𝜃 =

1

tan 𝜃

sin 𝜃 =1

csc 𝜃 cos 𝜃 =

1

sec 𝜃 tan 𝜃 =

1

cot 𝜃

Quotient Identities

tan 𝜃 =sin 𝜃

cos 𝜃 cot 𝜃 =

cos 𝜃

sin 𝜃

Cofunction Identities

sin 𝜃 = cos (𝜋

2− 𝜃) cos 𝜃 = sin (

𝜋

2− 𝜃) tan 𝜃 = cot (

𝜋

2− 𝜃)

csc 𝜃 = sec (𝜋

2− 𝜃) sec 𝜃 = csc (

𝜋

2− 𝜃) cot 𝜃 = tan (

𝜋

2− 𝜃)

Trig Functions of any Angle 𝜃

sin 𝜃 = 𝑦

𝑟 cos 𝜃 =

𝑥

𝑟 tan 𝜃 =

𝑦

𝑥 (𝑥 ≠ 0)

csc 𝜃 = 𝑟

𝑦 (𝑦 ≠ 0) sec 𝜃 =

𝑟

𝑥 (𝑥 ≠ 0) cot 𝜃 =

𝑥

𝑦 (𝑦 ≠ 0)

Chapter 6

Pythagorean Identities

sin2 𝑥 + cos2 𝑥 = 1 tan2 𝑥 + 1 = sec2 𝑥 1 + cot2 𝑥 = csc2 𝑥

Even-Odd Identities

sin(−𝑥) = − sin 𝑥 cos(−𝑥) = cos 𝑥 tan(−𝑥) = − tan 𝑥

csc(−𝑥) = − csc 𝑥 sec(−𝑥) = sec 𝑥 cot(−𝑥) = − cot 𝑥

b

a c

θ

θ (x, y) r

y

x

Sum and Difference Formulas*

cos(𝑢 − 𝑣) = cos 𝑢 cos 𝑣 + sin 𝑢 sin 𝑣

cos(𝑢 + 𝑣) = cos 𝑢 cos 𝑣 − sin 𝑢 sin 𝑣

sin(𝑢 − 𝑣) = sin 𝑢 cos 𝑣 − cos 𝑢 sin 𝑣

sin(𝑢 + 𝑣) = sin 𝑢 cos 𝑣 + cos 𝑢 sin 𝑣

tan(𝑢 − 𝑣) = tan 𝑢−tan 𝑣

1+tan 𝑢 tan 𝑣

tan(𝑢 + 𝑣) = tan 𝑢+tan 𝑣

1−tan 𝑢 tan 𝑣

Double-Angle Formulas*

sin 2𝑥 = 2 sin 𝑥 cos 𝑥

cos 2𝑥 = cos2 𝑥 − sin2 𝑥 cos 2𝑥 = 1 − 2 sin2 𝑥 cos 2𝑥 = 2 cos2 𝑥 − 1

tan 2𝑥 = 2 tan 𝑥

1−tan2 𝑥

Half-Angle Formulas*

sin𝜃

2= ±√

1−cos 𝜃

2

cos𝜃

2= ±√

1+cos 𝜃

2

tan𝜃

2= ±√

1−cos 𝜃

1+cos 𝜃 tan

𝜃

2=

sin 𝜃

1+cos 𝜃 tan

𝜃

2=

1−cos 𝜃

sin 𝜃

* memorization of tangent identities is at the discretion of the instructor

Chapter 7

Law of Sines

sin 𝐴

𝑎=

sin 𝐵

𝑏=

sin 𝐶

𝑐

Law of Cosines

𝑎2 = 𝑏2 + 𝑐2 − 2𝑏𝑐 cos 𝐴 𝑏2 = 𝑎2 + 𝑐2 − 2𝑎𝑐 cos 𝐵 𝑐2 = 𝑎2 + 𝑏2 − 2𝑎𝑏 cos 𝐶

Area of SAS Triangles

𝐾 =1

2𝑏𝑐 sin 𝐴 𝐾 =

1

2𝑎𝑐 sin 𝐵 𝐾 =

1

2𝑎𝑏 sin 𝐶

Area of SSS Triangles (Heron’s Formula)

𝐾 = √𝑠(𝑠 − 𝑎)(𝑠 − 𝑏)(𝑠 − 𝑐), where 𝑠 =1

2(𝑎 + 𝑏 + 𝑐) is the semiperimeter.

Vectors 𝒗 =< 𝑣1, 𝑣2 > and 𝒘 =< 𝑤1, 𝑤2 >

Magnitude ‖𝑣‖ = √𝑣12 + 𝑣2

2

Direction 𝜃 with cos 𝜃 = 𝑣1

‖𝑣‖, sin 𝜃 =

𝑣2

‖𝑣‖, and tan 𝜃 =

𝑣2

𝑣1

Dot Product 𝒗 ∙ 𝒘 = 𝑣1𝑤1 + 𝑣2𝑤2

Vector projection of 𝒗 onto 𝒘 𝑝𝑟𝑜𝑗𝒘𝒗 = 𝒗∙𝒘

‖𝒘‖𝟐 𝒘

Scalar projection of 𝒗 onto 𝒘 is 𝒗∙𝒘

‖𝒘‖

Work done 𝑊 = 𝑭 ∙ 𝑫

Polar Coordinates

𝑥 = 𝑟 cos 𝜃 𝑦 = 𝑟 sin 𝜃

𝑟 = √𝑥2 + 𝑦2 tan 𝜃 = 𝑦/𝑥

Polar Form of Complex Numbers 𝑧 = 𝑎 + 𝑏𝑖, 𝑧1 = 𝑟1(cos 𝜃1 + 𝑖 sin 𝜃1), 𝑧2 = 𝑟2(cos 𝜃2 + 𝑖 sin 𝜃2)

Modulus 𝑟 = |𝑧| = √𝑎2 + 𝑏2

Argument 𝜃 such that tan 𝜃 = 𝑏/𝑎

Product rule 𝑧1𝑧2 = 𝑟1𝑟2[cos(𝜃1 + 𝜃2) + 𝑖 sin(𝜃1 + 𝜃2)]

Quotient rule 𝑧1

𝑧2=

𝑟1

𝑟2 [cos(𝜃1 − 𝜃2) + 𝑖 sin(𝜃1 − 𝜃2)], 𝑧2 ≠ 0

Other Formulas- Formulas that students are not required to memorize, but are required to be able to use

in this course.

Chapter 6

Power-Reducing Formulas

sin2 𝑥 = 1−cos 2𝑥

2

cos2 𝑥 = 1+cos 2𝑥

2

tan2 𝑥 = 1−cos 2𝑥

1+cos 2𝑥

Product-to-Sum Formulas

cos 𝑥 cos 𝑦 =1

2[cos(𝑥 − 𝑦) + cos(𝑥 + 𝑦)]

sin 𝑥 sin 𝑦 =1

2[cos(𝑥 − 𝑦) − cos(𝑥 + 𝑦)]

sin 𝑥 cos 𝑦 =1

2[sin(𝑥 + 𝑦) + sin(𝑥 − 𝑦)]

cos 𝑥 sin 𝑦 =1

2[sin(𝑥 + 𝑦) − sin(𝑥 − 𝑦)]

Sum-to-Product Formulas

cos 𝑥 + cos 𝑦 = 2 cos (𝑥+𝑦

2) cos (

𝑥−𝑦

2)

cos 𝑥 − cos 𝑦 = −2 sin (𝑥+𝑦

2) sin (

𝑥−𝑦

2)

sin 𝑥 + sin 𝑦 = 2 sin (𝑥+𝑦

2) cos (

𝑥−𝑦

2)

sin 𝑥 − sin 𝑦 = 2 sin (𝑥−𝑦

2) cos (

𝑥+𝑦

2)

On the next page, the department has prepared a formula sheet that instructors may allow students to use

on Exam Two and the Final Exam.

FORMULA SHEET FOR MAT 1570 EXAM 3 AND FINAL EXAM

Addition & Subtraction Formulas for Tangent

tan(𝑠 + 𝑡) =tan 𝑠 + tan 𝑡

1 − tan 𝑠 tan 𝑡 tan(𝑠 − 𝑡) =

tan 𝑠 − tan 𝑡

1 + tan 𝑠 tan 𝑡

Double-Angle Formula for Tangent

tan 2𝑥 =2 tan 𝑥

1 − tan2 𝑥

Formulas for Lowering Powers

sin2 𝑥 =1 − cos 2𝑥

2 cos2 𝑥 =

1 + cos 2𝑥

2

tan2 𝑥 =1 − cos 2𝑥

1 + cos 2𝑥

Half-Angle Formulas for Tangent

tan𝑢

2=

1 − cos 𝑢

sin 𝑢=

sin 𝑢

1 + cos 𝑢

Product-to-Sum Formulas

sin 𝑢 cos 𝑣 =1

2[sin(u + v) + sin(u − v)]

cos 𝑢 sin 𝑣 =1

2[sin(u + v) − sin(u − v)]

cos 𝑢 cos 𝑣 =1

2[cos(u + v) + cos(u − v)]

sin 𝑢 sin 𝑣 =1

2[cos(u − v) − cos(u + v)]

Sum-to-Product Formulas

sin 𝑥 + sin 𝑦 = 2 sin𝑥+𝑦

2cos

𝑥−𝑦

2 cos 𝑥 + cos y = 2 cos

x+y

2cos

x−y

2

sin 𝑥 − sin 𝑦 = 2 cos𝑥+𝑦

2sin

𝑥−𝑦

2 cos 𝑥 − cos y = −2 sin

x+y

2sin

x−y

2

college algebra and trigonometry, ratti and … · mat 1570 – trigonometry classes meeting three...

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