LESSON 1: Trigonometry Pre-test

Transcription

1 LESSON 1: Trigonometry Pre-test Instructions. Answer each question to the best of your ability. If there is more than one answer, put both/all answers down. Try to answer each question, but if there is a question you do not know anything about, it is okay to write that you don t know. 1. What is sin (x)? 2. What is cos (x)? 3. What is tan (x)? 4. What is the purpose of the trigonometric functions? In other words, what types of problems can they help you solve? 5. Do sin (x) and cos (x) have any relationship? 6. How does sin x change as x goes from 0 to 90 degrees?

2 7. Are y = sin x and y = cos (x) functions? If so, what is their domain and range? 8. The height of a building s shadow is 56 ft when the sun is shining at a 35 angle to the horizon. What is the height of the building? Explain how you found your answer. h 56 ft 35

3 Notes to the teacher: Do not try to prepare students for the pre-test. It is okay if they don t know much. This is just to get a baseline to help us know what they understand going into this curriculum. The remainder of the day will be spent reviewing SOHCAHTOA and what they already should know about trigonometry, as well as introducing the basics of geometer s sketchpad. Things to know about GeoGebra: On the toolbar on the top, you can construct segments, circles, points, polygons, perpendicular bisectors, intersections, and take measurements. Each tool has a drop-down menu to do different, but similar actions. Students should spend some time exploring and getting comfortable with GeoGebra so that they can hit the ground running tomorrow. Make sure when they construct their shapes that they not only look right, but also that they can drag and move the applicable points in the appropriate manner. Sometimes they can make things look right, but they have constructed them incorrectly, and when they drag the point around, you will be able to tell. Students should construct the following: a point a segment a line a ray a triangle perpendicular lines the intersection point of two lines parallel lines a circle

4 LESSON 2: The goal of the first activity is to create a right triangle with hypotenuse 1 in GeoGebra, where we can move the triangle around any way we want, and find where the trigonometric functions are located on that triangle. 1. Open a new GeoGebra geometry window. 2. Use the segment tool to create a segment. Then use the measuring tool to measure that segment. Click on the measuring tool, select distance, and then click on the segment you want to measure. Adjust the measurement until it is 1cm. Try to make this segment as horizontal as possible. 3. Tip: Make sure you select the selection tool (looks like a pointer) in order to adjust the measurement. 4. Then use the circle tool to draw a circle using this segment as the radius. At this time, you might want to zoom in somewhat. 5. Construct an additional radius at something less than a 90 angle counterclockwise from the radius you already have. 6. Construct a perpendicular between the endpoint of your new radius and your horizontal segment. Click on the perpendicular line tool and the click the endpoint of the new radius as well as the horizontal segment. 7. Then construct the intersection point of the perpendicular line and the horizontal line. Click on the point tool and then hover over the intersection so that both lines are highlighted. At this point, click, and the new point will be constructed. 8. Highlight the perpendicular line, and hide it by going to Edit, and then Show/hide objects. 9. Draw a segment where the perpendicular line was, between the point that was on the circumference of the circle and the point that was found using the perpendicular line. 10. Draw a segment that goes along the base of the triangle, just up to where the perpendicular line intersected the x-axis. 11. Highlight the circle, right click, and select Hide circle. Highlight the line segment that is horizontal that is longer than the triangle, right click, and select, Hide segment. 12. Now you have a right triangle with radius 1, and as you change the angle (through the first quadrant, the right triangle is always a right triangle, and the hypotenuse is always Highlight the points of the triangle, beginning with the one that is in the origin, and continuing in a counter-clockwise fashion. Go to Edit, then Show/Hide Labels and then begin with A, then B, then C. Then, click on the fourth point, not in the triangle, and label it D. It is important that we all label our points the same to avoid confusion in the future. This is what your document should look like at this point.

5 14. Let s call the central angle x (this is angle CAB). 15. What is sin(x)? Using SOHCAHTOA, what would the ratio be? Knowing that the hypotenuse is 1, what does that tell you? 16. For which line segment is the length equal to sin(x)? Using the measurement tool, measure that line segment. If the measurement comes up in an inconvenient location, you can move it, but first you must click back to the selection tool. 17. What is cos(x)? Using SOHCAHTOA, what would the ratio be? Knowing that the hypotenuse is 1, what does that tell you? 18. For which line segment is the length equal to cos(x)? Using the measurement tool, measure that line segment. 19. Move the central angle so that x is very close to 0. What is sin(x) approximately? 20. Move the central angle so that x is close to 45. What is sin(x) approximately? 21. Move the central angle so that x is close to 90. What is sin(x) approximately? 22. Move the central angle so that x is very close to 0. What is cos(x) approximately?

6 23. Move the central angle so that x is close to 45. What is cos(x) approximately? 24. Move the central angle so that x is close to 90. What is cos(x) approximately? 25. Use the measurement tool to measure angle x. Select angle in the measurement tool. Then select segment AB and then AC (in that order). This will measure angle x. Remember, using the selection tool, you can move the measurement if it shows up in an inconvenient location. 26. This gives us a measurement of the height of the triangle in our coordinate plane. Remember that this height of the triangle is equal to sin(x) as we saw earlier. Move point C and see how this measurement changes. Write down a pattern that you see. 27. I don t want you to take my word for it. Put your calculator into degree mode, and type sin(x), using whatever the angle is for x into your calculator. Find sin(x). Is sin(x) equal to the height of the triangle? 28. Move the central angle so that x is very close to 0. Use your calculator to find sin(x) exactly. How does this compare to the measurement of the segment that is equal to sin(x)? 29. Move the central angle so that x is close to 45. What is sin(x) exactly? How does what your calculator says compare to the measurement from GeoGebra? 30. Move the central angle so that x is close to 90. What is sin(x) exactly? How does what your calculator says compare to the measurement from GeoGebra?

7 31. Next, move the angle to an angle around 60. Calculate cos(x) using your calculator. Is cos(x) equal to the length of the triangle? 32. Move the central angle so that x is very close to 0. What is cos(x) exactly? How does what your calculator says compare to the measurement from GeoGebra? 33. Move the central angle so that x is close to 45. What is cos(x) exactly? How does what your calculator says compare to the measurement from GeoGebra? 34. Move the central angle so that x is close to 90. What is cos(x) exactly? How does what your calculator says compare to the measurement from GeoGebra? 35. Now, go to Edit and then Show/Hide Objects. Highlight the perpendicular line through CB and hide it again. 36. Highlight AD and the point D, and got to Construct Perpendicular Line. Do you know what this line is called with respect to the circle? 37. As you may have remembered, that line is called a tangent line. 38. Construct ray AC (to do this, you can use the segment tool on the toolbar, but hold it down until you can select the ray tool). Then construct the intersection point of the tangent line and ray AC and label the intersection point E. 39. At this point, your document should look like this. (Note that you can drag your labels so that they are not covered up by your lines.)

8 40. Next, hide the tangent line and the ray AC, and after they are hidden, draw segments AE and DE. 41. The Ancient Greeks were the first to discover trigonometry, and they considered segment DE to be the tangent of angle x. Knowing that tangent is equal to opposite/adjacent in triangle ABC, can you show that segment DE is equal to tan(x) using the properties of similar triangles? 42. Using similar triangles, find the relationship between sin(x), cos(x), and tan(x). 43. Another way to look at the segment DE is by looking at triangle ADE, and consider finding the tan(x). What is the length of the adjacent segment in that triangle? What does that make tan(x) equal to?

9 44. Move the central angle so that x is very close to 0. What is tan(x) approximately? 45. Move the central angle so that x is close to 45. What is tan(x) approximately? Measure the segment DE. 46. Move the central angle so that x is close to 90. What is tan(x)? 47. Now use your calculator to find a value for tan(x). How do they compare? 48. Move the central angle so that x is very close to 0. Find tan(x) exactly in your calculator? How do they compare? 49. Move the central angle so that x is close to 45. What is the measurement of segment DE? What is tan(x) in your calculator? How do they compare? 50. Move the central angle so that x is close to 90. What is the measurement of segment DE? What is tan(x) in your calculator? How do they compare? 51. Save your document. We will keep using this as we continue to explore the trigonometric functions. HOMEWORK: Go home and ask your parents or grandparents how they did trigonometry. Did they use trigonometry tables? If so, do they remember what they were and how they worked?

10 LESSON 3: The goal of this activity is to create a trigonometry table for sin(x). Introduction: If you asked your parents or grandparents for homework last night, when they solved trigonometric equations, they probably used a table of values, rather than a calculator. In fact, when a calculator gives a value for a trigonometric equation, it is using a table of values that has been entered into its hard disk memory, in the same way that your calculator has a memory of the value of π to a certain number of digits. Your calculator does not know that π is the ratio of a circle s circumference to its diameter, and neither does it know anything about trigonometry, but rather, it simply has a trigonometry table in its memory. In ancient times, trigonometry tables were created by drawing a large and extremely precise circle, and measuring the lengths of the segments of sine at different angles. It was extremely time-consuming, difficult, and tedious. We are going to work together and use the measurement tool of GeoGebra to create a trigonometry table of our own. Teacher Note: Assign each student or pair of students to each whole-number degree value between 0 and 90, so that in the end, each value is covered twice. In your final trigonometry table, if two values disagree slightly because students used slightly different approximations of an angle, average them for the final table. If two values disagree significantly, investigate whether one student may have an error. You may want to set up a shared Google spreadsheet so that students can put their values into the spreadsheet, which can automatically average the values. Keep in mind, you will need to look over the student values to be sure there aren t errors. 1. In your GeoGebra document, go to Options, then Rounding, and change to 4 decimal places. 2. Zoom in as much as possible. 3. Your teacher will assign you several whole-number angle measurements. Move point C so that x is as close to each angle measurement as possible. If you can t get it exactly, get as close as possible, and record the measurement GeoGebra gives you for the length of CB. 4. All students should put their measurements up on the board, averages will be calculated, and everyone will record the final trig table on their own paper.

11 Angle (deg) Student 1 Student 2 Sin(x) Avg Angle (deg) Student 1 Student 2 Sin(x) Avg

12 LESSON 4: The goal of this activity is to use your trigonometry table to solve different problems involving trigonometry. Note that because trig tables are so difficult to create, you have to be creative with how you use them to make the most of the values they give you. If a value you are looking for is not given to you directly by your trig table, look at your GeoGebra document or go to and see if you can figure out a value that would be the same that is in your trig table. Teacher note: For numbers 1c-e and 4c-e, students will likely need help. Additionally, since GeoGebra in some cases is taking a measurement of distance, it won t measure when the value would be negative, while the calculator and common sense might tell us that we ought to have a negative answer. These will be good opportunities for discussion. This GeoGebra document: shows a similar picture to what students have created on their own, but with the trigonometric functions all appearing on the unit circle. It may be additionally helpful. 1. Using your trig table, find the following: Check using calculator: a. sin (47 ) b. sin (28 ) c. sin (150 ) (sketch a diagram that shows how you figured out which angle to use in your trig table) d. sin (97 ) (sketch a diagram that shows how you figured out which angle to use in your trig table) e. sin (-22 ) (sketch a diagram that shows how you figured out which angle to use in your trig table)

13 2. Using your trig table, solve for x in the following triangles: a. 17.5in 4.6in x Check using calculator: b. 4.3cm x 5.6cm Check using calculator: Side note: We already talked about the name of tangent. The word the Ancient Greeks used for sine meant chord, because sine is equal to half of the chord, but when it was translated from Greek to Arabic to Latin, there was a mis-translation that led to the word sinus in Latin, which means inlet being used, rather than the word that means chord.

14 The word cosine means sine of the complement because the cosine is actually the segment that is equal to the sine of (90-x). In the diagram below, we think of cosine as OM, but the Ancient Greeks thought of it as NP (which is congruent to OM). Remember that complementary angles add to 90. Sine has the same relationship to the original angle as cosine does to the complementary angle (90-x). Try turning the circle sideways to see this better. cosine sine 3. Even though our trigonometry table does not include cosine, how can we use this information to find cos(x) using our trig table?

15 4. Using your trig table, find the following: Check using calculator: a. cos (7 ) b. cos (88 ) c. cos (135 ) (sketch a diagram that shows how you figured out which angle to use in your trig table) d. cos (111 ) (sketch a diagram that shows how you figured out which angle to use in your trig table) e. cos (-59 ) (sketch a diagram that shows how you figured out which angle to use in your trig table)

16 5. Using your trig table, solve for x in the following triangle: 1.5in 0.6in x Check using calculator: 6. Now that you know how to find sin(x) and cos(x) using your trigonometry table, how can you find values for tan(x) using the trigonometry table? 7. Find the following using your trig table: a. tan (32 ) Check using calculator:

17 b. tan (83 ) Check using calculator: 8. When you have used your trig table and checked using your calculator, how close have your calculations been? When have they been off, and by how much? What do you attribute this difference to? Is this a lot of error, or just a little bit? 9. I am sure you will be happy to go back to using your calculator after this, but what have you learned from creating and using this trigonometry table? As you continue to use your calculator to solve a trigonometric equation, try to remember what is happening inside of your calculator.

18 LESSON 5: The goal of this activity is to define three new trigonometric functions, and use GeoGebra to draw the graphs of the functions as the angle x moves. 1. Secant is a trigonometric function that is less commonly used than sine, cosine and tangent. You may or may not have heard of it before. The Ancient Greeks considered sec(x) to be the segment AE. (Note: In geometry, a secant line refers to a line that intersects a circle in two places. If you extend the segment AE through both sides of the circle, it would be a secant line.) 2. Using the Perpendicular line tool, construct a line perpendicular to AB through A. Do this by clicking the tool, then click segment AB, then click point A. Then, use the point tool to construct the intersection point between the circle and the perpendicular line you just constructed. Label this point F. 3. Next select the vertical line you just created and point F, and construct a line perpendicular to the vertical line that goes through point F. 4. Then, draw ray AE, and construct the intersection between the most recent perpendicular line you created and ray AE. Label this intersection point G, and then hide the two perpendicular lines and ray AE. 5. Finally, construct segment AG and FG. Your document should look like this. 6. Now consider the segment FG. This is one of the trigonometric functions of the complementary angle. Which trigonometric function is it? 7. Since the sine of the complement is called cosine, what do you think FG should be called?

19 8. Now consider the segment AG. This is one of the trigonometric functions of the complementary angle. Which trigonometric function is it? 9. Since the sine of the complement is called cosine, what do you think AG should be called? Big new idea: Until this point, we have been using degrees to measure the angle x, but now we are going to switch and use something called radians. A radian measures an angle by how many radius distances the arc of the angle passes through. Since the distance around a whole circle is 2π(length of the radius), 360 = 2π 6.28 radians. This means that 180 = π 3.14 radians, and 90 = π/ radians. 10. Now we are going to convert our GeoGebra document to radians. Go to Options Advanced in Angle Units, change it to radians. 11. Look at your measurements. Drag angle x around the circle and notice what the radian measure is at different locations. Does this make sense using the conversions above? 12. If you ever want to refer back to this document, here is webpage with a similar document that has some additional features: It will be handy for future reference. 13. Next, we are going to graph the trigonometric functions. In order to create a graph, you are going to use a GeoGebra worksheet that is very similar to what you created, but has some extra features. 14. Go to the following link: To graph sin(x), check the box marked sine, and move the point around the circle to change the degree measurement of x (here called a). 16. You can also create the graph by clicking Start Animation.. You can click Erase Traces if you want to start over and make a new graph.

20 17. Use the graph of sin(x) to answer the following questions: a. What is the domain of sin(x)? Are there angles beyond what is shown in our graph? Are these acceptable angles for the domain? b. What is the range of sin(x)? c. Where is sin(x) positive, and where is it negative? 18. Now graph cos (x), after deleting the traces of sin(x). To graph cos(x), uncheck the box that says sine and check the box that says cosine. Either animate or move the point to graph cos(x). What segment on the circle corresponds with the height of the graph in this case? a. What is the range of cos(x)? b. Where is cos(x) positive, and where is it negative?

21 19. Go the following worksheet and graph tan(x). What segment on the circle corresponds with the height of the graph in this case? a. What is the range of tan(x)? b. Where is tan(x) positive, and where is it negative? c. Are there any angles where tan(x) is undefined? 20. Now erase all previous traces, and graph cot(x). What segment on the circle corresponds with the height of the graph in this case? a. What is its domain? Range? b. Where is cot(x) positive, and where is it negative? Is it ever 0? Is it ever undefined? 21. Now go to the following worksheet and graph sec(x). What segment on the circle corresponds with the height of the graph in this case? a. What is its domain? Range? b. Where is sec(x) positive, and where is it negative? Is it ever 0? Is it ever undefined?

22 22. Now erase all previous traces, and graph csc(x). What segment on the circle corresponds with the height of the graph in this case? a. What is its domain? Range? b. Where is csc(x) positive, and where is it negative? Is it ever 0? Is it ever undefined? Challenge/Extension: In the coming days, we will spend more time exploring how the 6 trigonometric functions are related to each other. Until then, see if you can answer any of these challenge questions (you may want to use GeoGebra to see if you can determine some of the answers): 1. Are there any pairs of trigonometric functions that are inversely related (that is, when one gets bigger, the other one gets smaller)? 2. Are there any pairs of trigonometric functions that are directly related (that is, they both get bigger together and smaller together)? 3. Three special cases of trigonometric functions are when they are equal to 0, equal to 1, or undefined. Is there any relationship among the trigonometric functions as to when that happens to which ones?

23 LESSON 6: The goal of this activity is to investigate the relationships between sine and cosine, tangent and cotangent, and secant and cosecant. You will graph them in pairs and determine how they are related to each other. Teacher note: It will be important to tease out why sin(x) and cos(x) and csc(x) and sec(x) are horizontal shifts of each other while tan(x) and cot(x) also require a reflection 1. Go to the following link: 2. Click both cosine and sine to graph both graphs at the same time. 3. Click Start Animation to create the graphs. 4. Which color represents sin(x) and which one represents cos(x)? How do you know? 5. What is the relationship between the graphs of sin(x) and cos(x)? 6. How can you incorporate a horizontal shift into a function? 7. Can you write cos(x) as a sin(x) function with a horizontal shift? 8. How does that make sense with what you know about the relationship between sin(x) and cos(x)? What about the words sine and cosine? How are those words related? Does that relate to the function you wrote in number 8? 9. Now go to graph tan(x) and cot(x) at the same time. 10. Which color represents tan(x) and which color represents cot(x)? 11. What is the relationship of tan(x) to cot(x)?

24 12. Can you write cot(x) as tan(x) with a horizontal shift? What else needs to happen besides a horizontal shift in this case? 13. How does that make sense with what you know about the relationship between tan(x) and cot(x)? What about the words tangent and cotangent? How are those words related? Does that relate to the function you wrote in number 12? 14. Go to and graph sec(x) and csc(x) at the same time. 15. Which color represents sec(x) and which color represents csc(x)? How do you know? 16. What is the relationship of sec(x) to csc(x)? 17. Can you write csc(x) as sec(x) with a horizontal shift? 18. How does that make sense with what you know about the relationship between sec(x) and csc(x)? What about the words secant and cosecant? How are those words related? Does that relate to the function you wrote in number 16? 19. If these functions are just horizontal shifts of each other, do we really need separate functions, or would it be sufficient to just have sin(x), sec(x) and tan(x)? 20. When is it helpful to have cos(x), cot(x), and csc(x)? 21. Are there ever cases where it seems redundant to have these additional functions? 22. See if your graphing calculator has a sine regression. Does it also have a cosine regression? Why do you think this would be?

25 LESSON 7: The goal of this activity is to investigate the relationships of the trigonometric functions on the circle. Introduction: For several days now, we have been working with a circle, whose radius is 1 unit. This circle is often called the unit circle, because it is a circle with a unit radius. On the unit circle, we can find several different right triangles. 1. See what right triangles you can find. You should be able to find three different right triangles (note that there are two right triangles that are congruent, we can just consider one of those). 2. Go back to your saved GeoGebra document. Start with the right triangle ABC. Construct this right triangle in GeoGebra using the Polygon tool on the tool bar. Note that you will have to highlight all 3 points and then highlight the first point again to construct the polygon (A-B-C-A, for example). Note: you can get rid of any unwanted measurements by right-clicking and selecting hide label. 3. Consider this sides of this triangle. What trigonometric function represents the length of AB? BC? What is the length of AC? 4. Go to Options, then Rounding, and select 2 decimal places. 5. Since this is a right triangle, can you apply the Pythagorean theorem to those side lengths? What do you get?

26 6. Now let s consider this relationship in another way. Go to View and CAS. Now you have a calculator on the side of your document. Now, find the names of the segments AB and BC by right clicking their measurements. Mine are called i and n. Put the lefthand side of the Pythagorean theorem into the calculator. See if it equals the right-hand side of the equation you found. Note that if you hit the equals sign, it will give you an exact value (with alarming accuracy!) but if you hit the approximately equal sign, it will give you something more reasonable. Your screen should look something like this: 7. Once you have the calculation, move point C around, and see if the calculation changes or stays the same. (Hint: the approximate calculation should stay the same, but the exact calculation should change this is because what we are doing here is not perfect. Our radius of 1 is not exactly 1 if you go out to enough decimal places. That is the error you are seeing in the exact calculations.) TEACHER NOTE: This would be a good time to stop and come together as a group to make sure everyone has created the correct identity sin 2 (x)+cos 2 (x)=1, and that they have been able to correctly enter that calculation into GeoGebra.

27 8. Delete triangle ABC, and now consider the triangle ADE. Construct the triangle with the polygon tool. 9. Consider this sides of this triangle. What trigonometric function represents the length of AE? DE? What is the length of AD? 10. Since this is a right triangle, can you apply the Pythagorean theorem to those side lengths? What do you get? 11. Now let s consider this relationship in another way. Go to the CAS and calculate the lefthand side of the equation you found. In a separate calculation, enter the right-hand side of the equation you found. See if the two sides are equal. Use approximate calculations. 12. Once you have the two calculations, move point C around. What happens to the two calculations? 13. Delete triangle ADE, and now consider the triangle AGF. Construct the triangle with the polygon tool. 14. Consider this sides of this triangle. What trigonometric function represents the length of AG? FG? What is the length of AF?

28 15. Since this is a right triangle, can you apply the Pythagorean theorem to those side lengths? What do you get? 16. Now let s consider this relationship in another way. Go to the CAS and calculate the lefthand side of the equation you found. In a separate calculation, enter the right-hand side of the equation you found. See if the two sides are equal. Use approximate calculations. 17. Once you have the two calculations, move point C around. What happens to the two calculations? 18. These three equations are known in trigonometry as the Pythagorean Identities. The first one is often called the principal Pythagorean identity. Does the name make sense? Why or why not?

29 LESSON 8: The goal of this activity is to investigate the relationships between sine and cosecant, cosine and secant, tangent and cotangent. It is also to investigate the relationship between among sine, cosine, and tangent. You will graph them in pairs/groups and determine how they are related to each other. 1. Go to Graph sin(x) and csc(x) at the same time. Which color represents sin(x) and which one represents csc(x)? 2. When is sin(x)=0? When is csc(x) undefined? 3. When sin(x) gets close to 0, what happens to csc(x)? 4. Try moving the slider for q and watching what happens to sine (BC) and cosecant (AG). Look at what happens when one gets very small, what happens to the other one. When one gets close to 1, what happens to the other one? 5. Make a table of values by sliding point C so that sin(x) is the following (round csc(x) to one decimal place): sin(x) csc(x)

30 6. Can you make any guesses as to the relationship between the graphs of sin(x) and csc(x)? TEACHER NOTE: You will have to stop here and discuss to make sure all students/groups have discovered the relationship. You may have to guide the class or have a group discussion to lead the class to the idea that they are reciprocal functions. 7. Erase the traces of sin(x) and csc(x), and graph cos(x) and sec(x) in different colors. Which color represents cos(x) and which one represents sec(x)? 8. When is cos(x)=0? When is sec(x) undefined? 9. When cos(x) gets close to 0, what happens to sec(x)? 10. Try moving the slider for q and watching what happens to cosine and secant. Look at what happens when one gets very small, what happens to the other one. When one gets close to 1, what happens to the other one?

31 11. Make a table of values by sliding point C so that cos(x) is the following (round sec(x) to one decimal place): cosx) sec(x) Can you make any guesses as to the relationship between the graphs of cos(x) and sec(x)? 13. Does this make sense based on the relationship you discovered earlier between sin(x) and csc(x)? 14. Go to and graph tan(x) and cot(x) at the same time. Which color represents tan(x) and which one represents cot(x)? 15. When is tan(x)=0? When is cot(x) undefined? 16. When tan(x) gets close to 0, what happens to cot(x)? 17. Try moving the slider for q and watching what happens to tangent and cotangent. Look at what happens when one gets very small, what happens to the other one. When one gets close to 1, what happens to the other one?

32 18. Make a table of values by sliding point C so that tan(x) is the following (round cot(x) to one decimal place): tan(x) cot(x) Can you make any guesses as to the relationship between the graphs of tan(x) and cot(x)? 20. Does this make sense based on the relationship you discovered earlier between sin(x) and csc(x) and cos(x) and sec(x)? 21. You already saw in Lesson 6 that tan(x) and cot(x) are horizontal shifts of each other. Can they have another relationship at the same time?

33 Challenge/Homework: Think back to Lesson 7, where we investigated the Pythagorean identities. 1. Many people use algebra to get from the principal Pythagorean identity to the other two Pythagorean identites. Try taking the principal Pythagorean identity, and dividing through by sin 2 (x). What do you get? 2. What can you divide by to get the final Pythagorean identity?

34 LESSON 9: In this lesson, the goal is to investigate what happens to the trigonometric functions when x arrives at certain special angles. 1. Go to Click on Special and Snap. This will allow you to see points where special trigonometric functions occur, and the angle will automatically snap to these angles. Notice that you can toggle between angles and radians. Now, GeoGebra is giving us all decimal measurements. Let s see if we can figure out the exact measurements for this triangle. 2. If sin(x) and cos (x) are the same, in other words, AB = BC. What is it called when a triangle has two sides with equal measures? 3. What does it mean for the angles when two sides of a triangle have equal measures? 4. If you know that the largest angle measure is 90 and the other two angles are the same degree measures, what would that make the degree measures of the other angles of this triangle? Draw it below.

35 5. What are these measures in radians? (Keep in terms of π.) 6. Find a decimal approximation for your answer above. Does that match with the decimal measure of x given in GeoGebra? 7. Using the Pythagorean theorem, find the lengths of the two missing sides of this triangle. Since you know that they are the same length, you can call them both x. Keep this answer exact. 1 x x 8. Convert the answer you got above to a decimal and compare it to the value that GeoGebra is giving you for sin(x) and cos(x). Is it the same?

36 9. This is a triangle that is known as a special right triangle. Triangles with these angles always have the same proportions, even when you scale up or scale down the size of the triangle. Using proportionality, find the following missing side lengths, assuming these are isosceles right triangles. 10. Now go back to GeoGebra and move point C so that sin(x) = Use the trigonometry table you created in Lesson 3 to find the degree measure of x at this time.

37 12. Use the formula you know (that π radians = 180 ) to convert this into radian measures. Then convert that into a decimal and compare it to the x you have in GeoGebra. Are they the same? If not, are they close? 13. If this is a right triangle, knowing one of the non-right angles, find the third angle. 14. Using the Pythagorean Theorem, knowing that the hypotenuse is 1 and the sin(x) side of the triangle is ½, find the third side of the triangle. 15. Fill in the diagram below with the missing side and angle measurements (use exact values, not decimals). 1 ½

38 16. This is another special right triangle. Triangles with these angles always have the same proportions, even when you scale up or scale down the size of the triangle. Using proportionality, find the following missing side lengths, assuming these are right triangles.

39 Extension: Using what you know about special right triangles, can you find the following points on the unit circle below (use exact values, not decimals). Go back to GeoGebra and examine it to see whether the sin(x) value would be the x-coordinate or the y-coordinate, and whether the cos(x) value would be the x-coordinate or the y- coordinate. Teacher note: Students may need help, especially getting started on this. You may want to do the first few points with them.

40 LESSON 10: Trigonometry Post-test Instructions. Answer each question to the best of your ability. If there is more than one answer, put both/all answers down. 1. What is sin (x)? 2. What is cos (x)? 3. What is tan (x)? 4. What is the purpose of the trigonometric functions? In other words, what types of problems can they help you solve? 5. Do sin (x) and cos (x) have any relationship? 6. How does sin x change as x goes from 0 to 90 degrees?

41 7. Are y = sin x and y = cos (x) functions? If so, what is their domain and range? 8. The height of a building s shadow is 56 ft when the sun is shining at a 35 angle to the horizon. What is the height of the building? Explain how you found your answer. h 56 ft On the following diagram, label anything that you can that is relevant to trigonometry, and explain how it is relevant.

42 10. If you apply the Pythagorean theorem to the triangles highlighted in each of the diagrams shown below, what trigonometric identity will you get? a. b.

43 11. Identify the following graphs, and explain how you know. c. d. e.

44 12. Write trigonometric functions that are equivalent to the following f = g. h. i. j = (4) = 123 (4) 561 (4) = 561 (4) 123 (4) = 13. In the following triangles, find the side lengths with exact measurements (do not use decimals). k l

45 Notes to the teacher: For the post-test, you will notice that some questions are identical to the pre-test. This is so that knowledge gains can be measured in a helpful way. Other questions have been added to the post-test to test on additional concepts they have learned during this unit. You may also notice that the test is very conceptual, and does not focus on testing skills much. To some extent, that is a reflection of the bridge curriculum s focus on concepts. Still, you may wish to add some questions that focus on testing skills. After the post-test, you may want to connect what you have been doing in this unit to what is coming next. There are various ways you might want to do this. You might use Lesson 9 as a jumping off point and map out the traditional unit circle, with all the special angles represented. You might use Lesson 7 as a jumping off point and discuss trigonometric identities. You might use Lessons 4, 5, and 8 as a jumping off point and discuss the graphs of the trigonometric functions and discuss all the possible transformations of those functions. You might use Lessons 3 and 4 as a jumping off point and solve problems using trigonometry. If you have used some or all of this curriculum, and you would like to share results, thoughts, or feedback, please contact me at j.r.vansickle@csuohio.edu.