Lesson Plan for Introducing Trigonometry
Goals
 For students to develop a sound understanding of the three basic trigonometric
functions.
 For the unit circle to become a reference that enables students to
calculate/estimate the values of trigonometric functions (this is preferable to trig
functions defined as ratios of sides in right triangles because it supports the full
domain for the functions).
 To introduce GeoGebra with a tool that students can turn to when they need to
envision the unit circle.
 Students need to know the common values of the trigonometric functions, and
reference to a unit circle enables the student to remember with understanding
what these values are; it is a goal that this be a start of that learning.
 To help the students switch from measuring angles in degrees to thinking in terms
of radians
Means
 Before class gets started students will have the unit circle file on their computers.
 At various points in the lesson, students will have to explain their thinking to
another student and evaluate another student’s arguments.
 As far as possible, the teacher will try to avoid her desire to jump in and “help”
the student by providing the answer!
Lesson
What follows is a rough script of a lesson. There are four investigations, or problem sets.
I will go over each section, have the Geogebra Circle up on a screen, and illustrate as I
am talking, and then hand out the written instructions for that section.
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I. GeoGebra Unit Circle
The GeoGebra Circle you have on your computer is a “Unit Circle”. What makes it so
special? The radius is equal to one, so the circumference is 2 п. An angle has been
created that lies in standard position (vertex at origin, one vector on the x-axis oriented
to the point (1,0) the second vector forming the other side of the angle). The slider
creates the potential values for the angle. Angles are measured as we travel
Created by Jean Lawlis (9/2008)
counterclockwise from the standard position. Take a minute to play with the slider for Ө
(theta). Notice that it can only go between 0 and 360 degrees. Imagine it can continue
around past 360 degrees (in trigonometry it can).
Where would your angle be “pointing” at 90 o? at 270 o ? At 360 o? At 390 o ?
 Can you envision how a negative angle would be represented? Write your
thoughts on a piece of paper.
 Describe on the same piece of paper how you could calculate a negative angle
using this tool?
 Trade papers with another student and explain your thinking to them.
 See if you agree on the answers to these questions: What positive angle would
correspond to - 90 o? -180 o ?
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
You can switch between radians and degrees by selecting options on the top tool bar and
then angle unit. We will be working in radians in Calculus, so switch to radian mode.
Each radian corresponds to one radius distance around the circumference starting from
standard position. There are 2 п radians in a 360 o angle. П is defined as the ration of
circumference to diameter of any circle.
 Can you explain why there are 2 п radians in a 360 o angle?
 Where would п radians be on the unit circle? What positive angle does this
correspond to in degrees?
 Trade your paper with another member of the class that you are not sitting with.
Read the other person’s explanations and decide if you agree.
 Where would п/4 radians be? What angle would it correspond to in degrees?
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Defining Sine, Cosine, and Tangent with the Unit Circle.
Sine, and cosine are functions that each take as an argument an angle measure (Ө) and
return numeric values. Given an angle in standard position on the unit circle, the cosine
will be the x coordinate of the terminating end of the angle, and the sine will be the y
coordinate. Your GeoGebra Unit circle shows the sine and cosine values on the left.
There is a dotted line that completes the right triangle made from the angle theta and the
point it terminates on the circle. Find the dotted line and notice how the triangle changes
as you change the value of Ө with the slider.
 What side of the triangle corresponds to cosine (Ө)? What side of the triangle
corresponds to sine(Ө)? What is the value of the hypotenuse? Does it change?
Created by Jean Lawlis (9/2008)
Why or why not?
 What are the maximum and minimum values of sine(Ө) and cosine(Ө)?
 A very important trig identity is based on this triangle. It states that the “sine of
theta, squared, plus the cosine of theta, squared, is equal to one.” Written:
sin2Ө + cos2Ө = 1
Can you explain with a sketch why this is true?
 If Ө =720o (or 4 п radians), what will sine (Ө) and cosine (Ө) be?
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Defining Tangent
The ratio of the sine(Ө) to the cosine(Ө) is defined to be the tangent(Ө).
 Looking at the triangle on your circle, Is it true that the hypotenuse of the triangle
has the slope tangent(Ө). Why or why not?
 What are the range of values the tangent can take?
 There is an actual line that is tangent to the unit circle at the point our angle
terminates on the circle. How could you use the tangent function to determine the
slope of that line?
 What is the equation of the line tangent to the circle at the point 1/4 the way
around the circle? 1/8th?
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Assignment and Looking Ahead
Make up your own questions about degrees/radians/trigonometric functions/the unit
circle to challenge your classmates. We will work using your questions to quiz each
other. Next class, we will look at using this tool to model the motion of a person on a
ferris wheel with a trigonometric function. This will become a future Calculus question
in which we will be able to answer questions like “When is the vertical velocity the
greatest for the rider, and how fast is he/she heading toward/away from the ground at
that point?” If a rider dropped an ice cream cone at that point, how fast would it be
going when it hit the ground?
Created by Jean Lawlis (9/2008)
Created by Jean Lawlis (9/2008)