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Thinking on the Unit Circle

Open 'Thinking on the Unit Circle' in a Separate, Resizable Window

Notes for the Teacher: One of the most commonly missed type of problems on my first exam in a trig class is problems of the following type:

Find θ where -180° < θ < -90° and sin θ = -0.3.

This program will help students figure out the answer accurate to the hundredths place. You can then ask them to calculate the answer to a greater degree of accuracy so you can be sure that the student figures out what to do in a calculator. The goal is to have the students use this program to figure out how to use the symmetries on the unit circle and their calculator's result of the inverse trig function to calculate the desired answer. After doing a few problems, the student should be able to answer questions of this type without the program. Instead they can use a piece of paper to draw a circle and figure the answer on their own. Of course when you get to the topic of composition of functions with their own inverse, you will want to revisit this program to show students again how to think about those kind of problems. Here are some things to discuss with your class:
  • Have your students think about which lines should be used to represent which trig functions: diagonal, horizontal, and vertical.
  • Ask your students to think about how they can use the symmetries of the unit circle to figure out their answers. In other words which angles are the same as the green angles and how do you use that to calculate the blue angle?
  • After you get to the topic of inverse trig functions, ask your students why the green dot is limited in its movement around the circle.
  • Ask your students how they can use the type of thinking illustrated in this program, even when they don't have the program in front of them.
  • Substitute the coordinates (a, b) and angle β for the numeric values and ask your students questions where they need to give their answers in terms of a, b, and β. Then they can just use the program to experiment, but not to get the answers.

Program Notes: The green dot has a limited range. The blue dot can be moved to the desired angle and will snap into where the green line crosses the circle if released near there. This program would make a great demonstartion tool in a classroom.

Last Update: Feb. 9, 2009