Portland Community College | Portland, Oregon

Lab time will be devoted primarily to small group activities emphasizing conceptual understanding and appropriate technology. The student’s role is to actively engage in positive collaboration with peers. Activities that can be used during lab are on the Mathematics Department home page at https://www.pcc.edu/programs/math/course-downloads.html

Upon completion of this course students should be able to:

1. Translate among various systems of measure for angles including radians, degrees, and revolutions.

2. Represent, manipulate, and evaluate trigonometric expressions in terms of sides of a right triangle and in terms of the coordinates of a unit circle.

3. Graph, transform, and analyze trigonometric functions using amplitude, shifts, symmetry, and periodicity.

4. Manipulate trigonometric expressions and prove trigonometric identities.

5. Solve trigonometric equations using inverses, periodicity, and identities.

6. Define, represent, and operate with vectors both geometrically and algebraically.

7. Apply the law of sines and the law of cosines to determine lengths and angles.

8. Use variables, trigonometric functions, and vectors to represent quantities, create models, find solutions, and communicate an interpretation of the results.

9. Determine the reasonableness and implications of mathematical methods, solutions, and approximations in context.

10. Demonstrate an understanding of the polar coordinate system including relevant functions.

Activities should 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. Activities will include group work.

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 \(0,\frac{\pi}{6},\frac{\pi}{4},\frac{\pi}{3},\frac{\pi}{2},\pi\), 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 50% 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. Graded homework
   2. Quizzes
   3. Group projects
   4. In-class activities
   5. Portfolios
   6. Individual projects/ written assignments
   7. Discussion Posts
5. Additional forms of assessment that do not have to be part of the grade:
   1. Attendance
   2. Individual student conference
   3. In-class participation
6. The Lab component will account for at least 15% of the grade, which will incorporate laboratory reports (graded work with an emphasis on proper notation and proper documentation) from students.

Course Topics

1. Angles
2. Periodic Functions
3. Right Triangle Trigonometry
4. Graphs of Sinusoidal Functions
5. Trigonometric Equations, Expressions and Identities
6. Oblique Triangle Trigonometry
7. Polar Coordinates and Vectors
8. Technology
9. Optional Topics

Course Content

1. Develop an understanding of angles in different systems of measure.
   1. Understand the definition of an angle in standard position and identify the initial and terminal rays.
   2. Express the measure of an angle in degrees, degrees-minutes-seconds (DMS), and radians.
      1. Convert between these different systems of measure.
   3. Sketch an angle of any given measure in standard position and identify the related or reference angle and coterminal angles.
   4. Find the length of an arc on the circumference of a circle
2. Explore and analyze periodic functions.
   1. Determine if a function is periodic.
   2. Determine the period of a periodic function.
   3. Determine the amplitude and midline of a periodic function where applicable.
   4. Define the sine and cosine functions in terms of the unit circle.
   5. Determine the period, midline, and amplitude of the sine and cosine functions.
   6. Define the tangent function in terms of the sine and cosine functions and determine its period.
   7. Define the reciprocal trigonometric functions.
3. Develop an understanding of right triangle trigonometry using both radians and degrees.
   1. Define the six trigonometric functions of an acute angle in terms of the sides of a right triangle.
   2. Solve right triangles:
      1. Given two sides.
      2. A side and a non-right angle of the triangle.
   3. Evaluate the exact values of the six trigonometric functions using \(30^{\circ}{-}60^{\circ}{-}90^{\circ}\) and \(45^{\circ}{-}45^{\circ}{-}90^{\circ}\) triangles.
   4. Model and solve applied problems involving right triangles and interpret in context.
4. Explore graphs of sinusoidal functions.
   1. Given graphs of sinusoidal functions, identify the amplitude, period, and midline and write an equation for the function. Some of the given graphs should require the use of horizontal shifts.
   2. Given equations of sinusoidal functions, identify the amplitude, period, and midline and draw the graph. Some of the given equations should involve horizontal shifts presented in factored and non-factored form.
   3. Investigate and express an understanding of the amplitude, period, and midline of a sinusoidal function in the context of applications.
5. Develop an understanding of, and skill in, solving trigonometric equations, simplifying trigonometric expressions, and verifying trigonometric identities.
   1. Simplify an expression using the fundamental identities (Pythagorean, reciprocal).

   2. Recognize and apply identities including the cofunction, sum and difference, double- and half-angle, and be made aware of the existence and usefulness of the product-to-sum and sum-to-product identities.

   3. Define the inverse trigonometric functions.

      1. Understand the domain and range restrictions, including the appropriate units.
      2. Understand how to use the inverse functions to find all solutions to a trigonometric equation.
   4. Find the general solution of a trigonometric equation symbolically and graphically, using exact values where appropriate.
   5. Find the solutions of trigonometric equations given domain constraints, using exact values where appropriate.
   6. Algebraically verify trigonometric identities.
   7. Distinguish between trigonometric identities which are always true and trigonometric equations which may or may not have solutions.
   8. Solve applied problems using trigonometry.
6. Demonstrate an understanding of solving problems using the Law of Cosines and the Law of Sines.
   1. Solve given triangles using the Law of Sines as appropriate; identify and solve the ambiguous case.
   2. Solve given triangles using the Law of Cosines as appropriate.
   3. Solve applications involving oblique triangles.
7. Demonstrate an understanding of polar coordinates and vectors and explore their use in real-world settings.
   1. Polar Coordinates
      1. Explore rose curves, lemniscates and limiçons using technology.
      2. Perform conversions between rectangular and polar coordinates.
      3. Plot points and simple graphs in polar coordinates.
      4. Define and graph the rectangular form of a complex number \(a+bi\) and polar form \(r e^{i \theta}\) and convert between them emphasizing Euler’s formula \(r e^{i \theta} = r \cdot \cos\left(\theta\right) + r \cdot \sin\left(\theta\right) \cdot i\)
   2. Vectors.
      1. Define a vector using magnitude and direction.
      2. Represent a vector in various forms, e.g., \(\vec{w}=3\hat{\imath}+4\hat{\jmath}=\left \langle 3,4 \right \rangle\).
      3. Apply vector operations of scalar multiplication, addition, and subtraction graphically and symbolically.
      4. Create unit vector in same direction as a given vector.
      5. Compute the dot product of two vectors.
         • Understand the significance of the sign of the dot product as it applies to the orientation of the vectors.
         • Find the angle between two vectors using the dot product.
      6. Investigate at least two of the following applications.
         • Tension in cables.
         • Work.
         • Component forces on objects.
         • Navigation.
         • Velocity vectors.
         • Other appropriate applied problems.
8. Use technology to enhance the understanding of concepts in the course.
   1. Select the appropriate mode for degrees and radians on the calculator.
   2. Conversion of fractions of a degree to minutes and seconds.
   3. Graph trigonometric equations in radian and degree modes in an appropriate viewing screen.
   4. Solve trigonometric equations graphically.
9. If time permits, the following topics are considered optional:
   1. Implicit Equations.
      1. Use circles and ellipses as examples of implicitly defined equations
   2. Parametric Equations.
      1. Use parametric equations to describe horizontal and vertical components of motion over time.
      2. Apply parametric equations to problems involving circular and elliptical motion, and/or parabolic trajectories.
      3. Write parameterizations of circles and ellipses.
      4. Using graphing technology to explore parametric equation.
   3. Complex Numbers
      1. Perform arithmetic operations on complex numbers.
      2. Use Euler’s formula to find an nth root of a complex number algebraically.

ADDENDUM

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: The following reference sheet may be provided to students during all exams and all other assessments. Items may be removed at an instructor’s discretion; however, nothing may be added.

https://spot.pcc.edu/math/download/112/mth112-reference.pdf