The Unit Circle is central to the study of Trigonometry in a Precalculus course. It is important that students learn all aspects of the unit circle very well for their success in precalculus and before continuing on to calculus. This project is designed to supplement a student's classroom instruction/reading with the various features of the unit circle to test the student's knowledge.
Click on each item for more details.
- A 'playground' will be provided where the teacher can demonstrate or the student can play with the features of the unit circle without the stress of being tested.
The background can be changed from rectangular to polar or a combination background. The key values can be marked with radio buttons that when clicked information pops up about that point on the circle. An angle marker can be dragged around the circle with a coordinate and an angle textbox updated with the movement.
- Be able to name a few standard angles in both radians and degrees when the terminal side is given. Also once given a standard angle, be able to locate the terminal side.
In part I: Random angles will be provided in 3 formats: degrees, radians in terms of π, and radians in terms of a rational number, then the student is to move the angle handle to the terminal side of the standard angle by dragging the angle handle. For part II: An angle will be displayed and the student is to give the standard angle as requested in one of the 3 formats described above.
- Given a point and one exact coordinate, the student should be able to calculate the other coordinate exactly or to a specified level of accuracy.
The point will appear on a unit circle image, so the student can see which quadrant it is in. A textbox will be available for the student to enter the answer with tools provided for a radical symbol as needed.
- Given a background rectangular grid with a point on the circle and a grid line, the student should be able to calculate both coordinates exactly.
One of the coordinates will be a simple fraction as can be determined by the grid scale. The other coordinate can be figured from the relationship: x2+y2=1. A textbox with tools to enter exact answers will be provided.
- Given an angle and its coordinates, be able to use symmetry to locate up to 7 other angles and the associated coordinates on the circle without a calculator.
Draggable points will be provided along with tools such as horizontal, diagonal, vertical, and perpendicular lines to help the student line up the points into the proper position.
- Use symmetry to answer a variety of questions about points on the unit circle in terms of a given angle β and given coordinates where P(β) = (a, b).
Questions will be one of the following types:
- Given one of the 7 symmetrical locations to the given angle, find an angle and a set of coordinates for that location. There are infinitely many possible angles so a function will be called to compare the terminal sides of the angles.
- Given a set of coordinates, find the location, and name an angle.
- Given an angle in terms of the provided angle, locate it on the unit circle and give the coordinates in terms of the provided coordinates.
- Test knowledge of the key values of the unit circle. The key values are the angles and their corresponding coordinates that students are expected to memorize or derive quickly, because they are easy to calculate.
The student will asked random questions in one of the following 3 formats:
- Given a random key angle, locate it and name the coordinates.
- Given a random set of coordinates, locate it and name a corresponding angle. Note that there are infinitely many possible angles so the program will need to check to see if the student's answer would land at the same terminal point.
- Given a random location, name the coordinates and a corresponding angle.
- Definition of Trig Functions based on the unit circle. Line segments can be drawn from an arbitrary standard angle to represent each of the 6 trigonmetry functions.
Either the line segments can be provided and the student idenfies which segment goes with which trig function by dragging trig functions to the segments or the student could be given a random trig function and angle and then asked to draw the line segment that represents the trig function for that angle. In this case the student could also be asked to calculate the trig functions value.
- Odd and even and cofunction relationships between and within trig functions can be figured from the unit circle and the definitions of the trig functions.
This concept will also be rolled into note 11 by composing a trig function with its invere cofunction. However, it is good to address this skill before learning about composition of functions since composition of functions is already difficult. The program will give a random trig function of an angle and ask which trig function of the complementary angle would produce the same answer. Then the student will illustrate both angles and draw in the lines that represent the answers to both trig functions. The student can then compare the lines and see that they are the same length. This visual should help them understand the concept.
- The unit circle can be used along with a calculator to find an angle in a specific interval that has a trig function with a specified value.
The student will be given a random trig function, its value, and an interval for the angle. The student can then calculate an angle that goes with the trig function and use the Unit Circle playground to help them figure out how to adjust the angle to get the corresponding angle in the desired given interval.
- When simplifying the composition of a trig function and an inverse trig function, the unit circle can also be used to find an exact answer.
The student will be given a random composition of functions problem with the inverse trig function on the outside and then the student can use the Unit Circle playground to help them find the exact answer. A possible extension of this problem would be to add problems where the inverse trig function is on the inside and then the student is to draw a right triangle in the proper quadrant and label the sides of the triangle to find exact answers. The student will need to learn to choose between the unit circle or the right triangle to solve the problems. Drawing and labeling tools will be provided.
- Eventually, the project should be made to be SCORM compliant so it should have an interface that can connect to a Learning Management System (LMS) to retrieve a student's name and record scores in the LMS gradebook.
SCORM stands for Shareable Content Object Reference Model. A SCO (Shareable Content Object) is a learning object that has been made to be SCORM compliant. Objects that are SCORM compliant can retrieve data from a LMS such as the user's name, previous score, and where they left off in the learning object. They can also record results in the LMS gradebook if the instructor sets them up to do so.
