Math 114 Information

MATH 114 INFORMATION Last updated June of 2016 by Kris Campbell and Abby Bailey

Text: College Trigonometry Version Corrected Edition by Carl Stitz and Jeff Zeager

This is an open source textbook available in pdf form for free. To minimize cost, we have trimmed the textbook down to only the required sections of material and sent this version to the bookstore. We have separated out the exercises. It is available as a separate purchase for students. We warned the bookstore to limit production of printed versions. That way if students do not choose to purchase a hard copy, the bookstore will not be left with a lot of excess material. So, please be aware that there may be limited availability of hard copies in the bookstore. Also, there may be a delay between a request from a student to purchase hard copies and the bookstore’s ability to provide the materials.

The trig homepage https://faculty.elgin.edu/dkernler/handbook/c/mth114.html includes the following • Single pdf of text for required sections (only) • Single pdf of homework exercises for required sections • Single pdf of homework answers for required sections • Individual pdf files by section of homework • Individual pdf files by section of answers • (In development) Geogebra files for images from the text • (In development) png files of images from the Geogebra files • (In development) power points • (In development) guided notes • (In development) sample assessment materials

Note: Student links should not go to this page.

We encourage you to share anything you might develop for this course by sending it to Dan Kernler, following the guidelines for submission to the department handbook.

You are, of course, free to make the full version available to your students as well and can download the pdf at http://www.stitz-zeager.com

Course Description: The primary objective of this course is to prepare students for calculus and post-calculus courses. Topics include analytical geometry, trigonometric functions, trigonometric identities, inverse trigonometric functions, and solving trigonometric equations.

Because trigonometry is a service course to the calculus sequence, all instructors were asked to provide feedback on the critical topics to cover. This information was gathered to ensure we properly prepare our students for the calculus sequence. The materials in development, the materials already created, and the information included here are based on that feedback. Mth114 was assessed during the Spring 2014 and Fall 2015 semester. The following recommendations came out of that assessment from a committee consisting of Terri Martin, Dan Kernler, Chalyce Deterding, and Der-Long Lin:

We recommended the following for future semesters in teaching MTH 114: • Instructors should prioritize memorization of reciprocal and quotient identities. • Instructors should prioritize memorization of at least the primary Pythagorean identity. • Instructors should prioritize memorization of the double angle identities. • Minimize/eliminate calculator use for determining values of trigonometric functions where the unit circle or right triangles can be used. • For graphing trigonometric functions, instructors are encouraged to teach multiple methods including “translations” as well as a “5 point table”.

Course Content Outcomes 1. Use the definition of radian measure to calculate exact and approximate values of arc length, area of a sector, linear velocity, angular velocity, and other circular relationships; 2. Recognize and use exact trigonometric function values for standard angles in radian and degree measures; 3. Solve triangles using geometry, law of sines, and/or law of cosines and use these techniques to solve applications; 4. Calculate areas of triangles using side lengths and/or angles; 5. Sketch graphs of trigonometric and inverse trigonometric functions; 6. Identify, use, and prove trigonometric identities by applying basic trigonometric identities and algebraic techniques; and 7. Solve trigonometric equations using algebraic techniques, trigonometric identities, and inverse trigonometric functions.

We have provided a rough correlation between the course outline and the sections that match up from the textbook. There also is a list of suggested problems for each of those sections.

Course Outline 1. Angle Measures § 10.1 a. Geometric relationships involving angles b. Similar triangles c. Radian and degree measures – Focus should be on radian measure, since it is the foundation for the derivative and integral formulas in calculus. Additionally, it was identified as crucial for all three semesters of calculus that students be able to evaluate all six trigonometric functions at all multiples of standard angles without the use of technology. So it is advised to emphasize this from the start. d. Applications of radians, including arc length, area of a sector, linear speed, and angular velocity

Suggested problems from 10.1: 1, 3, 4, 6, 7, 9, 10, 11, 12, 13, 14, 15-27 odds, 29-44, 50, 51, 53, 55, 56-62

2. Trigonometric Functions a. Definitions of the six trigonometric functions § 10.2, § 10.3 - Knowing the reciprocal identities of the trig functions is critical for all three semesters of calculus. Reverting back to sine and cosine definitions of the trig functions is critical for the material in the first two semesters of calculus. b. Calculating exact and approximate trigonometric values § 10.2, §10.3 - Focus should be on radian measure, since it the foundation for the derivative and integral formulas students will encounter in calculus. Additionally, it was identified as crucial for all three semesters of calculus that students be able to evaluate all six trigonometric functions at all multiples of standard angles without the use of technology. Yes, it is important enough to have us repeat it.

Suggested problems 10.2: 1-68, 69-72, 74 Suggested problems 10.3: 1-127, 133, 134, 135, 137

c. Graphing trigonometric functions § 10.5 – The ability to graph trig functions quickly and accurately without the use of technology was identified as a critical skill for all three semesters of calculus. a.i.Basic graphs in the Cartesian coordinate system a.ii. Translations of trigonometric functions a.iii. Periodicity, amplitude and phase shift

Suggested problems 10.5: 1-34, 38-43, (optional 44-51)

d. Applications § 11.2, § 11.3 a.i.Solving right triangles a.ii. Solving triangles using Law of Sines and Law of Cosines a.iii. Area of a triangle

Suggested problems 11.2: 1-23, 25-27, 31, 33, 35, 37 Suggested problems 11.3: 1-20,25,26

e. Further Applications, possibly including angles of elevation, bearing, and simple harmonic motion. § 11.1 – Simple harmonic motion is critical for students who are planning on engineering or physics majors. It shows up in differential equations, physics and engineering classes, and at an elementary level in second semester calculus.

Suggested problems 11.1: 1-12

3. Inverse Trigonometric Functions § 10.6 – This topic is critical for all semesters of calculus. Students need to understand the domain and range of the inverse trig functions as well as be able to graph them. They need know when they can “undo” a trig function with its inverse and when they can “undo” an inverse by applying the trig function (proper use of composition). a. Definitions of the inverses for the six trigonometric functions b. Calculating exact and approximate trigonometric values c. Graphing inverse trigonometric functions d. Applications

Suggested problems 10.6: 1-153 odds, 156, 157, 160, 161, 164, 165-183 odds, 185, 186, 188-210, 211, 213, 215 4. Trigonometric Identities § 10.3, § 10.4 – There was no consensus on which identities were the most important to have memorized (other than the Pythagorean identities). So students should be prepared for any range of expectations from their calculus instructors on what to have memorized. a. Definition of fundamental identities (including reciprocal, quotient, Pythagorean, and negative-angle identities) b. Verifying conditional statements and trigonometric identities – This process was identified as critical for all three semesters of calculus. c. Sum and difference identities d. Double and half angle formulas e. Applications (e.g. finding precise trigonometric values for angles other than 0, 30, 45, 60, 90 degrees) – This is the wording in curricunet, but we want to emphasize again that students should be thinking in radians rather than degrees for most of calculus.

Suggested problems 10.3: 1-127, 133, 134, 135, 137 Suggested problems 10.4: 1-48, 49-57 odds, 59-66, 71-91, 93, 94

5. Trigonometric Equations § 10.7 – This is critical in all three semesters of calculus. a. Finding solutions to trigonometric equations (possibly including solution restrictions) using: a.i.Algebraic techniques a.ii. Trigonometric identities a.iii. Inverse trigonometric functions

Suggested problems 10.7: 1-67 odds, 99-104

6. Optional Material – If time permits, the polar coordinate system and graphing in polar coordinates is far more important for the calculus sequence than complex numbers. These are in a separate file from the required text and exercises. These come into play at the end of second semester calculus and throughout the third semester. a. Polar coordinate system § 11.4 b. Graphing in polar coordinates § 11.5 c. Complex numbers in polar form, DeMoivre’s Theorem and roots of complex numbers

The applications that were identified as most important include: • solving right triangles • completely solving a triangle that does not have a right angle • gears • harmonic motion/modeling oscillatory behavior • bearing/navigation • law of sines If anyone develops a nice project for any of these, or has some favorite problems covering any of them, please share with the department by submitting it to Dan Kernler to be posted on the homepage. Concurrent Enrollment The ECC catalogue states “Concurrent enrollment in MTH 112 with consent of instructor”. This is only done in rare circumstances (typically if a student shows a strong math history but just misses placing into Mth133, with say a 27 on the ACT). If you are contacted by a student requesting to enroll in your course while also enrolling in Mth112, please contact a department coordinator first.

Tentative Schedule We have included a possible pace to follow for the material here. Week Topics Sections 1 Angle Measures in Degrees and Radians, Circular 10.1 Motion 2 Similar Triangles, Unit Circle Sine and Cosine 10.2 3 Right Triangle Trig 10.3 4 Fundamental Identities 10.3 5 Trigonometric Identities 10.3, 10.4 6 Trigonometric Identities 10.4 7 Trigonometric Identities 10.4 8 Graphing Sine and Cosine 10.5 9 Graphing Tangent and Cotangent 10.5 10 Graphing Secant and Cosecant 10.5 11 Inverse Trig and Trig Equations 10.6,10.7 12 Trigonometric Equations 10.7 13 Trigonometric Equations 10.7 14 Applications of Trig 11.1 15 Law of Sines 11.2 16 Law of Cosines 11.3 17