unit plan cover sheet – trigonometry lesson...

Unit Plan Cover Sheet – Trigonometry Lesson Plan

Name(s): Michael Clarke, Shelley Mourtgos, Kip Saunders, & Erin Shurtz Date: March 8, 2006Unit Title: Trigonometry Lesson PlanFundamental Mathematical Concepts (A discussion of these concepts and the relationships between these concepts that you would like students to understand through this unit).

STUDENTS WILL UNDERSTAND:

A circle can be divided into whatever size unit of angular measurement you would like and the basic trigonometric functions still work.

1) There are several systems of measurement of an angle. Although any size unit of angular measurement will work, some units are better for particular types of problems. 1-a) Degrees and Radians are the two most common units of angular measurement in which the full rotation corresponds to 360º and 2

radians.1-b) Gradians is another unit that has been developed in which the full rotation corresponds to 400 gradians (grads or gons) and a right

angle is 100 gradians (grads or gons).1-b-i) Gradians (a centesimal system) was first introduced by a German engineering unit to correspond to the circumference of the

earth (1 grad corresponded to 100 km of the earths 40,000 km circumference).

2) Angular units of measurement are arbitrary. Some units are more useful than others. The sine and cosine of 30º, 45º, and 60º yield irrational numbers. There are angles whose sine and cosine are rational.

3) The coordinates of the points we usually label on the unit circle come from special characteristics of equilateral and isosceles triangles.

4) Angle measurements we already know can be used to derive the trig identity for addition of two angles.

Describe how state core Standards, NCTM Standards, and course readings are reflected in this unit.

Course readings: Each day’s lesson is structured around an exploration task conducted in a small group setting, providing students with the opportunity to problem solve and communicate their ideas mathematically (Artzt & Armour-Thomas, Becoming a Reflective Mathematics Teacher, pp. 3-4

Each day’s lesson plan is structured to enhance classroom discourse by giving students the opportunity to discuss problems in small group settings prior to instructor interaction and input and then to move that discussion to the classroom setting (Artzt & Armour-Thomas,

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Becoming a Reflective Mathematics Teacher, pp. 16-18).

Students will discuss exploration principles as they work together in their small groups and will explain those principles as they present their findings to the class (Sherin, Mendez, & Louis, Talking about Math Talk, pp. 188-195).

This lesson plan strives to incorporate the “four faces of mathematics” by including opportunities for students to be creative as they compute, reason, and solve various problems as they come to know that a circle can be divided into a variety of angular measurements. The plan also seeks to have students comprehend the application of this understanding in daily life (Devlin, K. (2000). The four faces of mathematics. In M.J. Burke & F.R. Curcio (Eds.), Learning mathematics for a new century (2000 Yearbook). Reston, VA: National Council of Teachers of Mathematics).

Some specific places where Standards are addressed include: NCTM Standards:Problem Solving – build new mathematical knowledge through problem solving. Students will use what they already know about 30 and 45 degree angles to determine sine of 75 degrees.

Reasoning and Proof – make and investigate mathematical conjecturesAnalyzing the SG-3 scenario students will make conjectures that the information given must be an alternative form of measurement and through problem solving will gain an understanding that gradians are an alternative measurement of angles.

Students will conjecture about information they already know to determine and prove the addition identity.

Geometry – analyze characteristics and properties of two-and three-dimensional geometric shapes and develop mathematical arguments about geometric relationship; specify locations and describe spatial relationships using coordinate geometry and other representational systems; apply transformations and use symmetry to analyze mathematical situations; and use visualization, spatial reasoning, and geometric modeling to solve problems.

Students will analyze points within the unit circle based on drawing triangles to determine their coordinates and generalizing the geometric relationships that make this method of analysis work and using spatial reasoning.

Students will use special right triangles and geometric proofs to understand the addition identity.

Measurement – understand measurable attributes of objects and the units, systems, and processes of measurement; and apply appropriate techniques, tools, and formulas to determine measurements.

Students will identify what characteristics make a certain unit of angular measurement easy or hard to work with on the unit circle.

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Process Skills – Adapting the method of solving the problem as new elements are introduced in each new day’s task, based on reflection about the previous process of solving the problem and consolidating their mathematical thinking into statements that they can communicate with their peers (or teacher during the exploration and discussion stages.

Utah State Standards:Standard 2: Students will represent and analyze mathematical situations and properties using patterns, relations, functions, and algebraic symbols.Objective 2.3 Represent quantitative relationships using mathematical models and symbols.After finding coordinates for points based on the 3-4-5 triangle, students will develop a symbol and model to refer to the angles more concisely.

Standard 3: Students will solve problems using spatial and logical reasoning, applications of geometric principles, and modeling.Objective 3.1 Analyze characteristics and properties of two- and three-dimensional shapes and develop mathematical arguments about geometric relationships.Students will analyze properties of isosceles and equilateral triangles to develop relationships between the 30°, 45°, and 60° angles and their sines and cosines.Students will analyze characteristics of special right triangles to develop mathematical arguments about the sines of other angles.

Standard 4: Students will understand and apply measurement tools, formulas, and techniques.Objective 4.1 Understand measurable attributes of objects and the units, systems, and processes of measurement.Students will understand that grads are an alternate unit of angular measurement.

Resources:Downing, Douglas. (2001). Trigonometry the Easy Way. New York: Barron’s Educational Series, Inc.http://standards.nctm.org/document/appendix/process.htm

Outline of Unit Plan Sequence (Anticipation of the sequencing of the unit with explication of the logical or intuitive development over the course of the unit—i.e., How might the sequence you have planned meaningfully build understanding in your students?)

DAY 1: Exposure to gradians – an alternative way of measuring.A. Teacher presentation of SG-3 Scenario B. Student problem solving of angular measurement discrepanciesC. Student discovery of an alternative way of measuring that has 400 units in a circle.D. Explanation and discussion of gradians as a measurement system and the value and use of the system as an alterative

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http://standards.nctm.org/document/appendix/process.htm

measurement for angles.DAY 2: Finding rational points on the unit circle using special right triangles and the Pythagorean theorem

A. Take vote on whether or not there are more than 4 points with rational coordinates on the unit circle.B. Students explore to try to find more points.C. Students share methods for finding points. D. Discuss relationship between sine and cosine and the lengths of the right triangle.

DAY 3: Using rational points on the unit circle, develop new unit(s) of angular measurementA. Students work on worksheet to label other coordinates on the unit circle.B. Students share methods for finding points.C. Give a name to the base angle for the 3-4-5 triangle.D. Label the angles of the other points.

DAY 4: Ratios in degrees with the unit circle. Why we measure with the base we do. Where the values come from.A. Re-cap on the previous day. Other triangles make for “nasty” angles.B. What is the sin of 45 degrees? Why? Can you prove it? (classroom based discussion)C. In groups, derive the sin of 30 and 60 degrees.D. Many bases to choose from. Selected “easy “ one we could prove and work quickly with.

DAY 5: Angle measurements we already know can be used to derive the trig identity for addition of two angles. A. Review briefly previous lesson.B. Individual ideas of how to use special right triangles to represent sine of 75 degrees.C. Group work to find the answer.E. Group presentations on how they thought about solving the problem.F. Group discussion on how the geometric proof relates to the sine addition identity.

Tools (A list of needed manipulatives, technology, and supplies, with explanations as to why these are necessary and preferable to possible alternatives).

DAY 1: Stargate SG3 Overheads & handouts Overhead projector Each student needs calculator

DAY 2: White board and marker

DAY 3:

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Overhead & handouts of unit circles with different points based on (3/5, 4/5) Overhead project Each group needs calculator

DAY 4: White Board (to make initial presentation) Desks arranged in groups (to facilitate exploration)

DAY 5: White Board and marker Worksheets with 75 degree angle and right triangles drawn out.

DAY 1Exposure to gradians – an alternative way of measuring.

Unit Plan Sequence: Learning activities, tasks and

key questions (What you will do and say, what you will ask the students to do, how you will accommodate to unexpected

students’ mathematics)

Time Anticipated Student Thinking and Responses(What mathematics you will look for in student work and interactions.)

Formative Assessment

(to inform instruction and evaluate learning

in progress)Miscellaneous things

to rememberLaunching Student Inquiry

Put up overhead.

Ask a student to read the top paragraph.

Read out the calculations.

Pass out the student handouts.

Ask students: Are these the numbers you would expect

5 min. Overhead and handout say:You are a member of Stargate SG3 team and have gated to P3X797. You have encountered an alien device that appears to be from the Ancients. Daniel Jackson has translated one of the glyphs on the device to mean TRIGONOMETRY. Your job is to identify what trigonometric function this device calculates.

Students will not pay attention to the numbers at first.

Students will frantically start typing things into calculators:Some will notice: The values for 30 degrees and pi over 6 are not equal, etc.

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to rememberto get?What is different than what you would expect?

Some will notice: The values are closer for the degree measurements than for the radian measurements.

Supporting Productive Student Exploration of the Task (Students working in groups or individually)Tell students: Work together with your groups to try to figure out what this device is doing, what could be going on here.

15 min.

Some students: will try basic arithmetic variations on the sine or cosine function.

Some students: will switch back and forth between degrees and radians

Some students: will put the numbers into their calculator and try to get it to come up with a function.

Some students: will try using inverse trig. Functions to work backwards through the listed calculations.

Pay attention to which students have graphing calculators and which have scientific calculators.

Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)Ask each group what they did to try to get the numbers given.

Ask any group that figured out it was grads if they could determine how many grads make up a circle.

5 min. Students will somewhat envy any group that figured out that it was grads.

The group that figured it out will still feel a little frustrated, and like they were tricked. Most likely, they have a scientific calculator and will try to show the rest of the class how to change their calculator into grads.

They will describe how they used sines and cosines they had memorized to find their angles and set up ratios with the degree measurements to find that there are 400 total.

Ask each group for input. Include several different students in the discussion. Have one student come to the board and explain to the class how they arrived at the right conclusion.

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to rememberUnpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)

Ask what would be some of the benefits of a system that broke the circle into 400 angular units.

What system would you use? Why?

3 min. Some will answer: It’s easier to relate angles with the same sin/cosine in different quadrants.

Most will say that they like degrees better but will admit it was just because they learned it first.

Some will try to suggest that it is easier to divide 360 down into parts, but they will be unsure of themselves and might benefit from some class discussion of it.

Formative AssessmentExplain how the gradian system was developed by a German engineering unit to correspond to the circumference of the earth. Tell them that 1 grad corresponds to 100 km of the earths’ 40,000 km circumference.

Explain to the students that gradians are used most often in navigation and surveying (and infrequently in mathematics).

2 min. Students will synthesize that 100 km x 400 gradians (the circumference of the earth) = 40,000 km.

Ask the students to explain why this was a viable system for this purpose.

Ask the students why this would be the best system for these purposes.

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to remember

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DAY 2Finding rational points on the unit circle using special right triangles and the Pythagorean theorem.









Trace unit circle onto board.Put points at (±1, 0), (0, ±1).Say: Someone said for these points, both coordinates are rational numbers.

Ask: Do you agree?

Take vote: Are there any other points on this circle that have rational coordinates for both x and y?

Count: and write numbers on the board.

Say: We’re going to work in groups for a little while to try to figure this out.

5 min.

Most answer: YesA few: Have to think about it for a second

Some: will try to start reasoning out loud

Could be many, a few, or none.

If all think that there are others, tell them to prove it by finding as many of them as possible.

If all think that there aren't any others, tell them to prove it and try to come up with why that is.

If it's a good spread, tell them to explore the problem with their groups and try to come to a consensus as a group.

If too many have to think about it, ask: Can someone give us a quick reminder what rational means?

If too many are biasing others, say: just think to yourselves for a second

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to rememberSupporting Productive Student Exploration of the Task (Students working in groups or individually)

Ask: them: Can you think of any ways to manipulate that equation to make it easier to work with.

Ask them: Can you prove that those points are actually on the unit circle?

Tell them: See how many other points you can find.

15 min

Some: will try to use the equation x2 + y2 = 1 and try to find rational solutions, by solving, plugging in random fractions, or using their calculator.

Some: will start by writing down all the irrational points they have memorized and then stare blankly at their papers

Some: will instantly think of using a 3, 4, triangle normalized w\ hypotenuse of 1

Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)Say: Finish up your thought.

Say: Finish up your sentence.

5 min. If no one found any, have each group go through the methods they tried. Then, skip to lesson day 4 material on how we get the points for 30o, 45o, and 60o.

If a group found some, have them explain how they got them and write the points on the board. Make sure to draw a right triangle for one of the points. Have the other groups compare their processes with what that group did.

If they only found them by guess and check, have them prove that those points work in front of the class and see if anyone else in the class can use that proof as a springboard for developing a general method for finding rational points.

If still not quiet, say: Ok, finish up the word you were on.

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Emphasize the similarity between the Pythagorean theorem and the equation of a circle. Ask students to illustrate the relation.

5 min. They will start by being confused on where to put the right angle and then will realize that there are two possibilities, but neither is at the origin.

They will point out that the height, base length, and hypotenuse correspond to the sine, cosine, and radius of the point on the circle.

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DAY 3Starting with rational points on the Unit Circle, develop new units of Angular Measurement.









Ask: Last time we found that there were some points on the unit circle that had rational numbers for both coordinates. Remember?

Ask: Why did we have to do the division to get the points?

Introduce: Using the first point we found, to locate more rational points.

5 min. They’ll call out 3, 4, 5 and eventually also call out 3/5, 4/5.

Some will answer that the radius needed to be one.-- Point out that the radius corresponds to the hypotenuse of the triangle.

If not, ask if renaming angle measurements affects sine/cosine measurements any.

Supporting Productive Student Exploration of the Task (Students working in groups or individually)Pass out handouts and tell students to label the points.

Prod them to reduce it as much as possible.

Point them towards looking at the angle.

15 min.

At least one person in the group should be able to quickly complete the first two circles and night have to explain it to the other group members.

For the third circle, second point:- Some will think to find the angle and double it and then find sine and cosine.

They might think it is irrational.- Some will try to draw in more triangles connecting to the one they know and find

geometric relationships.

If a group is stuck, have them draw triangles and figure out which sides are the same length.

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to rememberPoint them to a calculator to get the exact angle.

Point them back towards looking at the angle and measuring it.

- Some might try to measure with a protractor or estimate visually. If they estimate x = 1/5 for second point, let them but make them find the y value and then have them prove it.

For the third circle, third point:- Some groups will use the same first method above.- Some will struggle to apply finding second point process to be able

to find third.To help reduce, ask if they notice any pattern in the fractions (denominators).

Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)Put up overhead.

Quickly have one student review the first two circles.

Have another group explain how they got values for the third circle.

After explanation ask if anyone else sees a faster way to do it.

5 min.

Will talk about negatives and about switching the x and y coordinates.

Will talk about finding the angle, doubling it, and using sine / cosine (or using the angle addition property).

Make sure to ask the rest of the class if they agree with what was labled.

If no one sees faster way, point back to triangles and lengths.

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Have class name the angle. Ask how many of that angle there are in a full circle.

Ask how many radians are in a full circle.

Re-emphasize: but how many is that?

Point out that the symbol lets us have an angular unit that doesn’t divide evenly into a circle.

Suggest 8M as such a symbol.

Go back and label the angles in circle 3, then 1, then 2.

Ask class what they think

5 min. Naming it with a word rather than a letter will confuse some.Some will estimate 10ish.

Some will grab calculator and answer exactly: 9.7640629 . . .

They will complain that it’s an annoying number.

They will all answer 2 pi, but will not realize that’s just as ugly of a number.

They will eventually answer about 6 or 6.28 (might take some prodding like “more that 10, less than 5?, etc.)

They will think it’s weird but slowly catch on as we label angles we’ve worked on today.

They will answer faster if you label 4M before trying 4M-1, for example.

Some will like the fact that we divided the circle into something we can’t actually

Use a word, not letter.

If not, ask if

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to rememberthis teaches about units of angular measurement and how we’ve picked them in the past.

measure and mention it.

Some will mention that it seems we can divide it however we want, with nothing being magical about the existing units.

Some will point out how angle names don’t affect sine and cosine

renaming angle measurements affects sine/cosine measurements any.

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DAY 4 Ratios in degrees with the unit circle. Why we measure with the base we do. Where the values come from.









Introduce Base-60 Wrap-up of day 3 What is the sine of 45? Why? Let’s prove it.

Why were we able to prove it?

5

5

Wondering why anybody would work in any of those bases. Start to wonder why we work with the base that we do for the unit circle.

Students will have enough of a background with trig that they will be able to answer the question. Some might even be able to explain why and a few may be able to prove it.

Students will see that it worked well because it was an isosceles triangle.

Have a student come and demonstrate why on the board.

Supporting Productive Student Exploration of the Task (Students working in groups or individually)What about ½ or (root 3)/2?Can you prove those?

10 Now without the isosceles triangles explore the other angles in the hope that they discover a way to make an equilateral triangle or some other clever way. It is anticipated that students will try to form a new shape out of two triangles like the example of the 45-45 triangle. Some might get caught with just one triangle. Some may make the wrong triangle, one that is isosceles but not equilateral.

Work individually and then in groups to find a method.

Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)Group Presentations 5 Choose a group who got it or got close to present on the board.

Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)Would any other angles allow for the proof? 5

Can we use this proof method for different sized triangles? Facilitate classroom discussion.

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DAY 4 Ratios in degrees with the unit circle. Why we measure with the base we do. Where the values come from.








to remember

So why the 30-60-90 triangle?

How can we estimate the sine of 70?

Students will see the special abilities we have with the 30, 60, and 45 degrees. Gain appreciation for choice of base. Now know why we break it up into those partitions. Specifically, that two 30-60-90 triangles can be combined to make an equilateral triangle of side 1. It is now possible to find the other lengths and get nice trig values. Also, the 45-45 is isosceles with a hypotenuse of 1, allowing us to solve the other sides. These give easy angles and easy lengths that break up the unit circle of 360 degrees into equal partitions.Use other angles we know to approximate 70 degrees. Leave it open for a discussion next class about other nice angles. i.e sin addition formula, etc…

Comprehension of where the numbers come from for the trig functions of the standard angles.

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DAY 5Angle measurements we already know can be used to derive the trig identity for addition of two angles









Review: “We’ve been talking about different ways to measure angles, and last time we talked about some particularly pretty angle measurements – what were they?”

Launch: Today we want to see if we can apply what we learned about those angles to find out information about other angles.

The problem we are going to be working with is “What is the sine of 75 degrees?”

Before you start working on that as a group, what are some different ways we can use what we know about the special right triangles to represent this problem?

7-10 min

Students will remember the special right triangles.

Some students will remember the addition identity (sin 75 = sin (45+30) = sin45*cos30+sin30*cos45). Some may remember to use the addition identity but not remember it correctly.

Some students will draw something that represents sin75 = sin (45+30). Others may set up something that represents sin75 = sin(30+45).

Others may set up a subtraction identity, such as sin75 = sin(90-15).

Others may set up the 75 degree angle “sideways” so that it isn’t on the unit circle.

Draw or have a student draw special right triangles and their side lengths.

Individual students will offer different ways of representing the problem and present them to the class.

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to rememberGive a hint: “ There are different ways of representing this problem and different ways for finding an answer. To get you started, I’m going to show you the easiest representation: If I extend lines of top triangle to form another right triangle (may erase other line) - does this still represent a 75 degree angle?”

“What are the unknown lengths we need to determine now to find sin(75)?” Draw outside little right triangle to give a simpler visual.

Students will hopefully see that this is the same angle.

Some students may want to use the triangle formed by extending the lines even further.

Some students may want to use the inside triangle rather than the outside triangle. If so, ask them what we can use to find out angles in the outside triangle rather than the inside one (linear pairs).

Students will identify opposite side and hypotenuse. They may need help seeing that they already know part of the other side and how the small outside triangle drawn can be used to find the missing length. They also may be confused about which hypotenuse they need to find.

Call on individual student to show/tell what the unknown lengths are.

Supporting Productive Student Exploration of the Task (Students working in groups or individually)

Hand out worksheets with 75 degree angle and right triangles already drawn. “Now work in your groups to see if you can use this

10 min

Students will work individually and in groups to find an answer.

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to rememberrepresentation to find the answer.”

Walk around and ask listen to group ideas. If they get stuck: help them see how to use linear pairs to find the angle measurements of the small outer triangles, ask them what they know about isosceles triangles, the Pythagorean theorem, or trig functions to find missing lengths.

Students may use any combination of trig functions, isosceles triangle theorem, Pythagorean theorem, and information about linear pairs to determine information about the missing hypotenuse and leg lengths.

Some students will not realize that they have an isosceles triangle with two legs having length 1 to work with.

Some students will find the missing adjacent side first and then use that and the Pythagorean theorem to find the missing opposite side.

Most students will not try to apply trig functions to the right triangles, but will mostly rely on the Pythagorean theorem to determine missing sides without paying attention to what is already known about the angles and their sines and cosines.

Some students may even introduce ideas about tangent and contangent.

Most students will probably not have enough time to finish the task.

Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)Have one group who got closest to the answer present their work. Let them answer any questions from other students.

If they did not get the answer, ask questions about

5 min Other students will be able to follow the groups reasoning and may have some questions as to why they did it the way they did.

Students may not have come to a final answer. One group may have come up with one of the missing sides and another group with the other missing side.

Group presentation

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to rememberisosceles triangles, linear pairs, the Pythagorean theorem, or trig functions to finish the problem as a class. Maybe have two groups present – one on finding the missing hypotenuse length and another on finding the missing leg length.

Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)“What relation can we see between this geometric proof and the sine angle sum identity?” Write on the board sin(75) = sin(30)cos(45) + cos(30)sin(45) = (y+1/2)/h

Conclusion: We can use what we know about special right triangles to find information about other angles and to understand the angle sum trig identities. Similar processes may be used to obtain other trig

5-7 min

Students may break up addition identity and try to find where the different components are shown in the geometric proof.

Individual student responses.

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to rememberidentities. Also if there is time, point out that these trig identities work no matter what angle measurement is used.

Summative AssessmentWe have discussed breaking the unit circle up into various units of angular measurement, including 400 grads, 2 radians, 360°, and angles utilizing the 3-4-5 Pythagorean Triple. We divide a day into 24 hours. Suppose we divide the unit circle into 24 “hours.” Determine the x and y coordinates for the following points and plot (and label) them on the unit circle:

(a) 2 “hours”(b) 5 “hours”(c) 9 “hours”

What are the advantages and disadvantages of using this system to divide up a circle?

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unit plan cover sheet – trigonometry lesson...

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