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# Elementary Functions

#### Prerequisites

MTH 111, RD 115, and WR 115, or equivalent placement

## Course Description

Investigates trigonometric functions, equations and identities. Examines right and oblique triangles, vectors, polar coordinates, parametric equations, and complex numbers. Explores topics graphically, numerically, symbolically, and verbally. Graphing calculator required. Prerequisite: MTH 111, RD 115, and WR 115, or equivalent placement. Audit available.

## Intended Outcomes

Upon successful completion students should be able to:

1. Analyze real world scenarios to recognize when trigonometric functions, vector arithmetic, the polar coordinate system, or parametric equations are appropriate, formulate problems about the scenarios, creatively model these scenarios (using technology, if appropriate) in order to solve the problems using multiple approaches, judge if the results are reasonable, and then interpret and clearly communicate the results.
2. Appreciate trigonometric, parametric, and vector concepts that are encountered in the real world, understand and be able to communicate the underlying mathematics involved to help another person gain insight into the situation.
3. Work with trigonometric functions, vector arithmetic, the polar coordinate system, and parametric equations in various situations and use correct mathematical terminology, notation, and symbolic processes in order to be prepared for future coursework in calculus and the sciences that requires the use of and an understanding of the concepts of elementary functions.

## Outcome Assessment Strategies

Assessment shall include:

1. The following must be assessed in a proctored, closed-book, no-note, and no-calculator setting: finding the exact values of the trigonometric functions at integer multiples of 0, π/6, π/4, π/3, π/2 and π, using the Pythagorean identities, and using the relationships between the six trigonometric functions.
2. At least two proctored, closed-book, no student-notes (an instructor-provided list of identities and formulas is allowed; see Addendum B) exams, one of which is a comprehensive final exam that is worth at least 25% of the overall grade. The proctored exams should be worth at least 60% of the overall grade. These exams must consist primarily of free response questions although a limited number of multiple choice and/or fill in the blank questions may be used where appropriate.
3. Various opportunities to express - and be graded on - mathematical concepts in writing. Assessment should be made on the basis of using correct mathematical syntax, appropriate use of the English language, and explanation of the mathematical concept.
4. At least two of the following additional measures:
1. Take-home examinations
3. Quizzes
4. Group projects
5. In-class activities
6. Portfolios
7. Individual projects
5. Additional forms of assessment that do not have to be part of the grade: (see math department web resource for additional strategies).
1. Attendance
2. Individual student conference
3. In-class participation

## Course Activities and Design

All activities will follow the premise that formal definitions and procedures evolve from the investigation of practical problems. In-class time is primarily activity/discussion emphasizing problem solving techniques. Activities will include group work.

## Course Content (Themes, Concepts, Issues and Skills)

#### 1.0 ANGLES

Develop an understanding of angles in different systems of measure.

1.1 Understand the definition of an angle in standard position and identify the initial and terminal rays.
1.2 Express the measure of an angle in degrees, degrees-minutes-seconds (DMS), and radians.

1.2.1 Convert between the angle-measures listed in 1.2.

1.3 Sketch an angle of any given measure in standard position and identify the related or reference angle and coterminal angles.
1.4 Find the length of an arc on the circumference of a circle using the definition of an angle in radian measure.

#### 2.0 PERIODIC FUNCTIONS

Explore and analyze periodic functions.

2.1 Determine if a function is periodic.
2.2 Determine the period of a periodic function.
2.3 Determine the amplitude and midline of a periodic function where applicable.
2.4 Define the sine and cosine functions in terms of the unit circle.
2.5 Determine the period, midline, and amplitude of the sine and cosine functions.
2.6 Define the tangent function in terms of the sine and cosine functions and determine its period.
2.7 Define the reciprocal trigonometric functions.

#### 3.0 RIGHT TRIANGLE TRIGONOMETRY

Develop an understanding of right triangle trigonometry using both radians and degrees.

3.1 Define the six trigonometric functions of an acute angle in terms of the sides of a right triangle.
3.2 Solve right triangles given two sides or a side and a non-right angle of the triangle.
3.3 Evaluate the exact values of the six trigonometric functions using 30°– 60°– 90° and 45°– 45°– 90° triangles.
3.4 Solve applied problems involving right triangles.

#### 4.0 TRANSFORMATIONS OF TRIGONOMETRIC FUNCTIONS

Explore and analyze transformations of trigonometric functions.

4.1 Investigate families of trigonometric functions within the context of transformations represented graphically, symbolically, numerically and verbally.

4.1.1 Shift trigonometric functions horizontally or vertically.
4.1.2 Reflect trigonometric functions about the horizontal or vertical axis.
4.1.3 Stretch trigonometric functions vertically and horizontally.
4.1.4 Given graphs of sinusoidal functions, identify the phase shift, horizontal shift, amplitude, period and midline and write an equation for the function.
4.1.5 Given equations of sinusoidal functions, identify the phase shift, horizontal shift, amplitude, period and midline and draw the graph.
4.1.6 Investigate and express understanding of given transformations in the context of applications.

4.2 Fit sinusoidal functions to data.

4.2.1 Fit sinusoidal functions to data analytically using the concepts of horizontal shift, amplitude, period and midline.
4.2.2 Investigate and express understanding of the models in the context of applications.

#### 5.0 TRIGONOMETRIC EQUATIONS, EXPRESSIONS AND IDENTITIES

Develop an understanding and skill in solving trigonometric equations symbolically and graphically in real world settings.

5.1 Simplify an expression using the fundamental identities (Pythagorean, reciprocal).
5.2 Recognize other identities such as the cofunction, sum and differences, double and half angle, product to sum identities.
5.3 Define the inverse trigonometric functions.

5.3.1 Understand the domain and range restrictions.
5.3.2 Understand how to use the inverse functions to find all solutions to a trigonometric equation.

5.4 Find the general solution of a trigonometric equation symbolically and graphically.
5.5 Find the solutions of trigonometric equations given domain constraints.
5.6 Find the exact solutions to equations involving trigonometric functions where appropriate.
5.7 Algebraically verify trigonometric identities.
5.8 Distinguish between trigonometric identities which are always true and trigonometric equations which may or may not have solutions.
5.9 Solve applied problems using trigonometry.

#### 6.0 OBLIQUE TRIANGLE TRIGONOMETRY

Develop an understanding and skill in solving problems using the Law of Cosines and the Law of Sines.

6.1 Solve given triangles using the Law of Cosines as appropriate.
6.2 Solve given triangles using the Law of Sines as appropriate; identify and solve the ambiguous case.
6.3 Solve applications involving oblique triangles.

#### 7.0 POLAR COORDINATES, VECTORS, and PARAMETRIC EQUATIONS

Gain skill in the use of polar coordinates, vectors and parametric equations and explore their use in real world settings.

7.1 Polar Coordinates.

7.1.1 Plot points and graphs in polar coordinates.
7.1.2 Perform conversions between rectangular and polar coordinates.
7.1.3 Explore rose curves, lemniscates and limiçons using technology.

7.2 Vectors.

7.2.1 Define a vector using magnitude and direction.
7.2.2 Convert a vector to component form.
7.2.3 Apply vector operations of scalar multiplication, addition, and subtraction graphically and symbolically.
7.2.4 Compute the dot product using both the component form and magnitude-angle form of the vectors.

7.2.4.1 Understand the significance of the sign of the dot product as it applies to the orientation of the vectors.
7.2.4.2 Find the angle between two vectors using the dot product.

7.2.5 Investigate at least two of the following applications.

i. Tension in cables.
ii. Work.
iii. Component forces on objects.
v. Velocity vectors.
vi. Other appropriate applied problems.

7.3 Parametric Equations.

7.3.1 Use parametric equations to describe horizontal and vertical components of motion over time.
7.3.2 Apply parametric equations to problems involving circular and elliptical motion, and/or parabolic trajectories.
7.3.3 Write parameterizations of circles and ellipses.

7.4 Implicit Equations.

7.4.1 Use circles and ellipses as examples of implicitly defined equations.

#### 8.0 COMPLEX NUMBERS

Develop an understanding and skill in solving problems involving operations on complex numbers.

8.1 Define a complex number and perform conversions between, and arithmetic operations on, rectangular and polar forms (Euler's formula: ).
8.2 Find an nth root of a complex number algebraically.

#### 9.0 TECHNOLOGY

Use technology to enhance the understanding of concepts in the course.

9.1 Select the appropriate mode for degrees and radians on the calculator.
9.2 Conversion of fractions of a degree to minutes and seconds.
9.3 Graphing parametric equations.
9.4 Graphing polar equations.
9.5 Graph trigonometric equations in radian and degree modes in appropriate windows.
9.6 Solve trigonometric equations graphically and numerically.

Documentation Standards for Mathematics
All work in this course will be evaluated for your ability to meet the following writing objectives as well as for "mathematical content".

1. Every solution must be written in such a way that the question that was asked is clear simply by reading the submitted solution.
2. Any table or graph that appears in the original problem must also appear somewhere in your solution.
3. All graphs that appear in your solution must contain axis names and scales. All graphs must be accompanied by a figure number and caption. When the graph is referenced in your written work, the reference must be by figure number. Additionally, graphs for applied problems must have units on each axis and the explicit meaning of each axis must be self-apparent either by the axis names or by the figure caption.
4. All tables that appear in your solution must have well defined column headings as well as an assigned table number accompanied by a brief caption (description). When the table is referenced in your written work, the reference must be by table number.
5. A brief introduction to the problem is almost always appropriate.
6. In applied problems, all variables and constants must be defined.
7. If you used the graph or table feature of your calculator in the problem solving process, you must include the graph or table in your written solution.
8. If you used some other non-trivial feature of your calculator (e.g., SOLVER), you must state this in your solution.
9. All (relevant) information given in the problem must be stated somewhere in your solution.
10. A sentence that orients the reader to the purpose of the mathematics should usually precede symbol pushing.
11. Your conclusion shall not be encased in a box, but rather stated at the end of your solution in complete sentence form.
12. Line up your equal signs vertically.
13. If work is word-processed all mathematical symbols must be generated with a math equation editor.

Identities and Formulas Reference Sheet: