Developing the Unit Circle
Lesson 7 of 13
Objective: SWBAT find the coordinates of special angles and their multiples on a unit circle.
An angle's terminal side passes through the point (-8, -3). Which trigonometric functions will have a value greater than 0?
I give students a few seconds to think. At first some students will say, "None." As I wait for more responses, others will eventually offer tangent as a possibility to which I will respond, "I was thinking of only sine and cosine, but that's a good suggestion?" I will then ask students to name all 6 trigonometric functions so that I can list them on the board. After writing all the functions, I will have student volunteers explain why tangent and cotangent are positive. I expect that my students will either say, "The opposite and adjacent sides are negative so when you divide you get positive," or "the x- and y-coordinates are both negative." On Page 2 I've included a sketch beneath the prompt. Many of my students will have used a diagram like this to help them interpret the question correctly.
Next, I will have my students give the ratios for all six trigonometric functions. We will reason in terms of right triangles and in terms of x, y and r. We will also discuss how the only quantities that have direction are x and y. r represents the radius which and is defined as a distance measurement, in other words it is a scalar.
Now that we have reviewed and extended the previous lesson's ideas, we are ready to look at special angles on the unit circle.
I now prompt students to remember some material from geometry. My students geenrally recall that a 45-45-90 triangle is an isosceles right triangle, so the legs are equal. When I ask "How can we find the hypotenuse?" most of my students will use the Pythagorean Theorem. I move students from using "x" for a side by saying "Let's use a convenient value for x." We label the triangle with the values on Page 2 of special triangle development.
Remembering the 30-60-90 triangle is often harder for my students, so I draw an equilateral triangle (pg. 3). I ask students the following questions to guide the review:
- What do you know about the sides and angles of an equilateral triangle? Let's make the sides equal 2.
- What happens if we drop and altitude? What happens to the angle? The base?
- How can we find the value of the altitude?
Now that we have the triangles I ask students to find the value of sin 60. At first students may look confused. One issue is students do not understand what is meant by finding the value. I discuss we want the output or in this case the ratio for sin 60. This helps students and they begin discussing using the triangle to find the value. A student puts the answer on the board and explains how to find the answer.
I now give students several questions that requires more reasoning. I do not give any hints but let the students discuss how to find the answers. As students work with tan 150, I hear 150 is 30 degrees from the x axis. After students have discussed the question. Ideas are shared out to the class. As you can see on page 2 students draw diagrams for each of the problems. Students draw a reference angle the a triangle and label the triangle to determine the value.
Now that students have an idea of what we are doing I move to use what they did to construct the unit circle.
Creating the unit circle
I begin the development of the unit circle by have students relabel the special triangles so the hypotenuse is 1. This is a great time to remind students of scale factors from geometry. Students are given The Unit Circle activity. The directions show students how to label their diagram. The one issue students have with the unit circle is that they do not understand that it is conventional to label the terminal side of the angle. Many students think that the line is, itself, the angle. I try to help my students achieve a better understanding by stating in the directions that we are labeling the terminal side of the angle. I will discuss with the class that the reason for the convention is to make the diagram easier for others to read. If we were to draw all the angles from the initial side to the terminal side it would get very messy and confusing.
I let students determine how they would like to find the coordinates. For those who I sense are struggling I will ask these questions to help them get their thinking on track:
- What do you need to find?
- How are the x and y related to trigonometric functions (remember what we did in the last couple of lessons)
- Do you know the sine and cosine for the angle?
- When you draw the triangles for the angles is the hypotenuse 1?
- If we drew an triangle with a hypotenuse as 1 would it be similar to the triangle you already know?
- How could you find the sides of a triangle with the hypotenuse as 1?
Some students will need more questioning than others. Once the students begin to see the relationships they will be able to find all the angles around the circle. I do keep extra unit circle graphs for students since many will make errors and want to make a final clean copy.
Once students have completed their unit circle diagram, I want them to take some time to analyze the result (MP7). I ask the following:
- Do you use any patterns as you were completing the unit circle?
- Do you see any patterns with the angles as you move around the circle?
- If you look at all the radian measures in quadrant II what do you notice about the angles? (some realize that the numerator is 1 less than the denominator)Why does this seem reasonable? (I do this with quadrant III and IV)
Seeing the patterns in the circle will help students understand how to reproduce the circle and visualize the circle when they are working with special angles.
As students finish their unit circle I ask them to find the value of a trigonometric function. Some students see how to use the circle while other will need some help.
I give students several more expressions to evaluate and move around the room to help. I also explain to students that they need to determine a method (using special triangles or using the unit circle) that is easy for them to understand and use. As I move around I hear many students discussing reference angles and using those to find the values. Some students are changing all the radian measures to degrees while others are working in radians.
As we move into later lessons and units students will become more confident with radians and will be able to convert special angles (multiplies of 30 and 45) relatively quickly.
As class ends I want to analyze student understanding. I ask students to answer the following question on the Exit Slip:
Explain how knowing the coordinates of the angles in the first quadrant and using reference angles will allow you to find the value of a trigonometric functions whose terminal side is in quadrants 2,3 and 4.
This question allows me to analyze students understanding of the structure of the unit circle.