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?
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
LESSON 2: The goal of the first activity is to create a right triangle with hypotenuse 1 in Geometer s Sketchpad, where we can move the triangle around any way we want, and find where the trigonometric functions are located on that triangle. 1. In your Geometer s Sketchpad sketch, go to Graph, Define Coordinate System. Then use the circle tool to draw a circle whose center is at 0 and whose radius is 1 (drag the outside of the circle to x=1). At this time, you probably want to zoom in. You can do this by selecting the point where the circle intersects the positive x-axis and dragging it to the right. 2. Then, use the segment tool create a segment that goes from the center of the circle out to the edge of the circle. This will allow you to change the segment any way you want while keeping it length 1. 3. Construct a segment that goes along the x-axis from the center of the circle to the edge of the circle. 4. Construct a perpendicular between the endpoint of your original segment and your segment that goes along the x-axis. Highlight that point and also highlight that segment. Go to Construct, Perpendicular Line. 5. Then highlight the segment that goes along the x-axis and the new perpendicular line. Go to Construct, Intersection. 6. Highlight the perpendicular line, and click hide. 7. 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. 8. Draw a segment that goes along the base of the triangle, just up to where the perpendicular line intersected the x-axis. 9. Highlight the circle, right click, and select Hide circle. Highlight the line segment that is along the x-axis that is longer than the triangle, right click, and select, Hide segment. 10. 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 1. 11. Highlight the points of the triangle, beginning with the one that is in the origin, and continuing in a counter-clockwise fashion. Go to Display, then Label points and then begin with A and click OK. 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 sketch should look like at this point. 12. Let s call the central angle x.
13. What is sin(x)? Using SOHCAHTOA, what would the ratio be? Knowing that the hypotenuse is 1, what does that tell you? 14. For which line segment is the length equal to sin(x)? 15. What is cos(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 cos(x)? 17. Move the central angle so that x is very close to 0. What is sin(x) approximately? 18. Move the central angle so that x is close to 45. What is sin(x) approximately? 19. Move the central angle so that x is close to 90. What is sin(x) approximately? 20. Move the central angle so that x is very close to 0. What is cos(x) approximately?
21. Move the central angle so that x is close to 45. What is cos(x) approximately? 22. Move the central angle so that x is close to 90. What is cos(x) approximately? 23. Highlight the three points that make up the angle x, in this order: BAC. Go to Measure, and select Angle. In the upper left corner of the screen, you should see a precise angle measurement. Highlight the measurement, right-click, go to Label, and enter x. 24. Next, to measure the height of the triangle, Geometer s Sketchpad can help. Unfortunately, Geometer s sketchpad only measures in cm, not in the units that we have created. Since we have enlarged our graph, one unit does not equal one centimeter. To account for that, we need to divide by the length of the hypotenuse. First, highlight the height of the triangle. Go to Measure, then Length. Then, highlight the hypotenuse. Go to Measure, then Length. Next, go to Number, then Calculate. Click on the measurement of the height, then the division symbol on the calculator, and then the measurement of the hypotenuse, and click OK. 25. 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. 26. Move the central angle so that x is very close to 0. What is sin(x) exactly? How does this compare to what you estimated in #17? 27. Move the central angle so that x is close to 45. What is sin(x) exactly? How does this compare to what you estimated in #18? 28. Move the central angle so that x is close to 90. What is sin(x) exactly? How does this compare to what you estimated in #19?
29. Geometer s sketchpad will measure sin(x) directly, instead of us having to measure the ratio. Let s compare this to the ratio we measured. Go to Number, then Calculate. Type sin() and inside the parentheses to put x, click on the measurement of x on the left-hand side of your sketch, and then click OK. Move the point C, and see how this number compares with the ratio that we measured. Is sin(x) equal to the height of the triangle? 30. Next, calculate cos(x). We know that we could measure it using a ratio, but for simplicity, we will calculate it, since we know it will be equal to the measurement. Go to Number, then Calculate. Type cos(x) and click OK. 31. Move the central angle so that x is very close to 0. What is cos(x) exactly? How does this compare to what you estimated in #20? 32. Move the central angle so that x is close to 45. What is cos(x) exactly? How does this compare to what you estimated in #21? 33. Move the central angle so that x is close to 90. What is cos(x) exactly? How does this compare to what you estimated in #22? 34. Now, go to Display and then Show all Hidden. Highlight the perpendicular line through CB and hide it again. 35. 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?
36. As you may have remembered, that line is called a tangent line. 37. Construct ray AC (to do this, you can use the segment tool on the left-side 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. 38. At this point, your sketch should look like this. (Note that you can drag your labels so that they are not covered up by your lines.) 39. Next, hide the tangent line and the ray AC, and after they are hidden, draw segments AE and DE. 40. 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? 41. Using similar triangles, find the relationship between sin(x), cos(x), and tan(x).
42. 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? 43. Move the central angle so that x is very close to 0. What is tan(x) approximately? 44. Move the central angle so that x is close to 45. What is tan(x) approximately? 45. Move the central angle so that x is close to 90. What is tan(x) approximately? 46. Go to Calculate, enter tan(x), and click OK. 47. Move the central angle so that x is very close to 0. What is tan(x) exactly? How does this compare to what you estimated in #43? 48. Move the central angle so that x is close to 45. What is tan(x) exactly? How does this compare to what you estimated in #44? 49. Move the central angle so that x is close to 90. What is tan(x) exactly? How does this compare to what you estimated in #45? 50. Save your sketch. 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?
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 Geometer s Sketchpad to create a trigonometry table of our own. 1. In your Geometer s Sketchpad sketch, right-click on the measurement below mcb mac =.67 then click value, and change Precision from hundredths to ten-thousandths. 2. Drag the point D to the right as much as possible to zoom in on your drawing. 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 Geometer s Sketchpad gives you for 567 586. 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.
Angle (deg) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 Student 1 Student 2 Sin(x) Avg Angle (deg) 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 Student 1 Student 2 Sin(x) Avg
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 Geometer s Sketchpad sketch, and see if you can figure out a value that would be the same that is in your trig table. 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)
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.
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?
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)
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:
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.
LESSON 5: The goal of this activity is to define three new trigonometric functions, and use Geometer s Sketchpad 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. Next highlight segment AB and point A, go to Construct, and select Perpendicular Line. Then, highlight the line you just created and the circle, go to Construct, and select Intersection. Label this point F. 3. Next select the line you just created and point F, go to Construct, and select Perpendicular Line. 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 sketch 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?
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/2 1.57 radians. 10. Now we are going to convert our Geometer s Sketchpad sketch to radians. Go to Edit Preferences in Units go to the drop down menu in Angle and change it to radians. 11. Look at your measurements. Delete any measurements of segments and ratios so that all you have left is the measure of x and four trigonometric functions. 12. Next, we are going to graph the trigonometric functions. In order to create a graph, you are going to use a Geometer s Sketchpad sketch that is very similar to your sketch, but it has a few extra features that will make graphing easier. 13. Open this new document called Bridge Curriculum Sketch for Students.gsp. It should look very similar to the sketch you were working on. 14. To graph sin(x), highlight the measurement of x and then the measurement of sin(x). Then go to Graph and Plot as (x,y). Then go to Display and Trace Plotted Point. Move point C around the circle slowly, and the graph of sin(x) will appear. Notice that the height of the point that is tracing the graph is the same as the height of the triangle, which is equal to sin(x), because what we are graphing is points of the form (x, sin(x)), so the y-value is equal to sin(x). 15. You can also create the graph by highlighting point C, going to Display, and Animate Point. If the graph appears spotty, try slowing down the animation. In Display, you can also erase the traces if you want to start over or make a new graph.
16. 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? 17. Now graph cos (x), after deleting the traces of sin(x). To graph cos(x), highlight the measurement of x and then the measurement of cos(x). Then go to Graph and Plot as (x,y). Then go to Display and Trace Plotted Point. Either animate or move point C 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?
18. Erase the traces of sin(x) and cos(x), 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? 19. Now erase all previous traces, 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? 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 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 your Geometer s Sketchpad sketch to play around and 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?
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. 1. Start by graphing sin(x). 2. Highlight the measurement of x and then the measurement of sin(x). Then go to Graph and Plot as (x,y). Then go to Display and Trace Plotted Point. Move point C around the circle slowly, and the graph of sin(x) will appear. Notice that the height of the point that is tracing the graph is the same as the height of the triangle, which is equal to sin(x), because what we are graphing is points of the form (x, sin(x)), so the y-value is equal to sin(x). 3. You can also create the graph by highlighting point C, going to Display, and Animate Point. If the graph appears spotty, try slowing down the animation. In Display, you can also erase the traces if you want to start over or make a new graph. 4. The next thing we will do is graph cos(x) in in a different color, leaving the graph of sin(x) in place. 5. Graph cos (x) using the same procedure. Do not delete the traces of sin(x). To graph cos(x), highlight the measurement of x and then the measurement of cos(x). Then go to Graph and Plot as (x,y). As soon as you do that, a point will appear. Right-click on the point, and change its color, so that the sin(x) trace will be a different color from the cos(x) trace. Then go to Display and Trace Plotted Point. Either animate or move point C to graph cos(x). 6. What is the relationship between the graphs of sin(x) and cos(x)? 7. How can you incorporate a horizontal shift into a function? 8. Can you write cos(x) as a sin(x) function with a horizontal shift? 9. 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?
10. Erase the traces of sin(x) and cos(x), and graph tan(x) and cot(x) in different colors. 11. What is the relationship of tan(x) to cot(x)? 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. Erase the traces of tan(x) and cot(x), and graph sec(x) and csc(x) in different colors. 15. What is the relationship of sec(x) to csc(x)? 16. Can you write csc(x) as sec(x) with a horizontal shift? 17. 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? 18. 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)? 19. When is it helpful to have cos(x), cot(x), and csc(x)? 20. Are there ever cases where it seems redundant to have these additional functions? 21. See if your graphing calculator has a sine regression. Does it also have a cosine regression? Why do you think this would be?
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. Start with the right triangle ABC. Construct this right triangle in geometer s sketchpad using the Polygon tool on the left side. 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). 3. Consider this sides of this triangle. What trigonometric function represents the length of AB? BC? What is the length of AC? 4. Since this is a right triangle, can you apply the Pythagorean theorem to those side lengths? What do you get?
5. Now let s consider this relationship in another way. Go to Number and Calculate. Then enter the left-hand side of the equation you have found (using the trigonometric functions). See if it equals the right-hand side of the equation you found. Note that you can click the measurements on the left-hand side of your sketch to make them appear in the Calculator. 6. Once you have the calculation, move point C around, and see if the calculation changes or stays the same. 7. Delete triangle ABC, and now consider the triangle ADE. Construct the triangle with the polygon tool. 8. Consider this sides of this triangle. What trigonometric function represents the length of AE? DE? What is the length of AD? 9. Since this is a right triangle, can you apply the Pythagorean theorem to those side lengths? What do you get? 10. Now let s consider this relationship in another way. Go to Number and Calculate. First enter the left-hand 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. 11. Once you have the two calculations, move point C around. What happens to the two calculations? 12. Delete triangle ADE, and now consider the triangle AGF. Construct the triangle with the polygon tool.
13. Consider this sides of this triangle. What trigonometric function represents the length of AG? FG? What is the length of AF? 14. Since this is a right triangle, can you apply the Pythagorean theorem to those side lengths? What do you get? 15. Now let s consider this relationship in another way. Go to Number and Calculate. First enter the left-hand 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. 16. Once you have the two calculations, move point C around. What happens to the two calculations? 17. 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?
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. Start by graphing sin(x). 2. Highlight the measurement of x and then the measurement of sin(x). Then go to Graph and Plot as (x,y). Then go to Display and Trace Plotted Point. Move point C around the circle slowly, and the graph of sin(x) will appear. Notice that the height of the point that is tracing the graph is the same as the height of the triangle, which is equal to sin(x), because what we are graphing is points of the form (x, sin(x)), so the y-value is equal to sin(x). 3. You can also create the graph by highlighting point C, going to Display, and Animate Point. If the graph appears spotty, try slowing down the animation. In Display, you can also erase the traces if you want to start over or make a new graph. 4. The next thing we will do is graph csc(x) in in a different color, leaving the graph of sin(x) in place. 5. Graph csc (x) using the same procedure. Do not delete the traces of sin(x). To graph csc(x), highlight the measurement of x and then the measurement of csc(x). Then go to Graph and Plot as (x,y). As soon as you do that, a point will appear. Right-click on the point, and change its color, so that the sin(x) trace will be a different color from the csc(x) trace. Then go to Display and Trace Plotted Point. Either animate or move point C to graph csc(x). 6. When is sin(x)=0? When is csc(x) undefined? 7. When sin(x) gets close to 0, what happens to csc(x)? 8. Try moving point C around the circle 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?
9. 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) -1-0.5 0 0.1 0.2 0.5 1 10. Can you make any guesses as to the relationship between the graphs of sin(x) and csc(x)? 11. Erase the traces of sin(x) and csc(x), and graph cos(x) and sec(x) in different colors. 12. When is cos(x)=0? When is sec(x) undefined? 13. When cos(x) gets close to 0, what happens to sec(x)?
14. Try moving point C around the circle and watching what happens to cosine (AB) and secant (AE). 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? 15. 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) -1-0.5 0 0.1 0.2 0.5 1 16. Can you make any guesses as to the relationship between the graphs of cos(x) and sec(x)? 17. Does this make sense based on the relationship you discovered earlier between sin(x) and csc(x)? 18. Erase the traces of cos(x) and sec(x), and graph tan(x) and cot(x) in different colors. 19. When is tan(x)=0? When is cot(x) undefined? 20. When tan(x) gets close to 0, what happens to cot(x)?
21. Try moving point C around the circle and watching what happens to tangent (DE) and cotangent (FG). 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? 22. 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) -1-0.5 0 0.1 0.2 0.5 1 23. Can you make any guesses as to the relationship between the graphs of tan(x) and cot(x)? 24. Does this make sense based on the relationship you discovered earlier between sin(x) and csc(x) and cos(x) and sec(x)? 25. 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?
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?
LESSON 9: In this lesson, the goal is to investigate what happens to the trigonometric functions when x arrives at certain special angles. 1. Open the sketch titled Bridge Curriculum Sketch for Students. Move point C so that as close as possible sin(x) = cos(x). It should look like this. Now, Geometer s Sketchpad 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.
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 Geometer s Sketchpad? 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 Geometer s Sketchpad is giving you for sin(x) and cos(x). Is it the same?
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 your Geometer s Sketchpad document and move point C so that sin(x) =.5 (or as close as you can possibly get to this). 11. Use the trigonometry table you created in Lesson 3 to find the degree measure of x at this time.
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 Geometer s Sketchpad. 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
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 30-60-90 right triangles.
Extension: Using what you know about special right triangles, can you find the following points on the unit circle below. Go back to your Geometer s Sketchpad file 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.
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?
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 9. On the following diagram, label anything that you can that is relevant to trigonometry, and explain how it is relevant.
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.
11. Identify the following graphs, and explain how you know. c. d. e.
12. Write trigonometric functions that are equivalent to the following f. 9 :;< = = g. h. i. j. 9 >?@ = = 9 :AB (=) = :;< (=) >?: (=) = >?: (=) :;< (=) = 13. In the following triangles, find the side lengths with exact measurements (do not use decimals). k. 4.5 60 90 30 l. 45 8 90