RIGONOMERY 4.. 4..6 In the first part of Chapter 4, students consider different slope triangles for a given line or segment and notice that on each line, the slope remains constant no matter where the draw the slope triangle on that line or how large or small each slope triangle is. hese slope triangles allow students to find lengths of sides and angle measures that the previousl could not calculate. his stud leads to the tangent relationship, the first of the trigonometric functions we will stud. Using the tangent function and their calculators, students are able to find measurements in application problems. See the Math Notes boes on pages 90, 94, and 200. Eample he line graphed at right passes through the origin. Draw in three different slope triangles. For each triangle, what is the slope ratio,!!? What is true about all three ratios? Note:! and! are read change in and change in. slope triangle is a right triangle that has its hpotenuse on the line that contains it. his means that the two legs of the right triangle are parallel to the aes: one leg runs verticall, the other horizontall. here are infinitel man slope triangles that we can draw, but it is alwas easiest if we draw triangles that have their vertices on lattice points (that is, their vertices have integer coordinates). he length of the horizontal leg we call and the length of the vertical leg we call. t right are three possibilities. For the smallest triangle, = 3 (the length of the horizontal leg), and = 2 (the length of the vertical leg). For the smallest triangle we have!! = 2 3. 36 Geometr Connections Parent Guide
In the medium sized triangle, = 6 and = 4, which means:!! = 4 6 Lastl, we can find the same lengths on the largest triangle: = and =, so:!! = If we reduce the ratios to their lowest terms we find that the slope ratios, no matter where we draw the slope triangles for this line, are all equal.!! = 2 3 = 4 6 = he students also discovered that when the slope of the line (and thus the hpotenuse) changes, the slope ratio changes as well: the steeper the line, the larger the slope ratio, and the flatter the line, the smaller the slope ratio. Eample 2 Using the tangent button on our calculator rather than our rig able resource page, find the missing side length in each triangle. a) b) q 62 9.6 20 w 22 When using the tangent button on a calculator with these problems, ou must be sure that the calculator is in degree mode and not radian mode. Your student should be able to check this and fi it, if necessar. Since we found that the slope ratio depends on the angle, we can use the angle measure and the tangent button on the calculator to find unknown lengths of the triangle. In part (a), we know that the tangent of the angle is the ratio opposite!leg =! adjacent!leg!. his allows us to write the equation at right and solve it. We found the value of tan 62 on the calculator. tan 62 = q opposite leg 9.6 ( ) adjacent!leg 9.6(tan 62 ) = q q! (9.6)(.88)! 8.048 Chapter 4: rigonometr and Probabilit 37
In part (b) we will set up another equation similar to the previous one. his equation is slightl different from the one in our first eample in that the variable is in the denominator rather than the numerator. his adds another step to the solution process. Some students might realize that the can rotate the triangle and use the 70 angle (which the would have to determine using the sum of the measures of the angles of the triangle) so that the unknown side length is in the numerator. tan 20 = 22 w w = 22 tan 20 w! 60.44 Eample 3 alula is standing 7 feet from the base of the Washington Monument in Washington, D.C. She uses her clinometer to measure the angle of elevation to the top of the monument to be 78. If alula s ee height is feet, 3 inches, what is the height of the Washington Monument? With all real world applications, the first step is alwas the same: draw a picture of what the problem is describing. ere, we have a girl looking up at the top of the monument. We know how far awa alula is standing from the monument, we know her ee height, and we know the angle of elevation of her line of sight. We translate this information from the picture to a line diagram, as shown at right. On this diagram we include all the measurements we know. hen we write an equation using the tangent function and solve for..2 ft 78 7 ft We add the ee height to the value of to find the height of the Washington Monument: 49.9 +.2 =. feet. tan 78 = 7 7(tan 78 ) =! 49.9 feet 38 Geometr Connections Parent Guide
Problems For each line, draw in several slope triangles. Calculate the slope ratio.. 2. 3. 4. Chapter 4: rigonometr and Probabilit 39
Use the tangent button on our calculator to solve for the variables. It ma be helpful to rotate the triangle so that it resembles a slope triangle.. 6. 4 z m 3.2! 70 28 7. 8. 2 33 c 48 89 80 Careful!! 9..! w 47 2.2 4. ladder makes a 7 angle with the wall it is leaning against. he base of the ladder is feet from the wall. ow high up the wall does the ladder reach? 2. Davis and ess are 30 feet apart when ess lets go of her helium-filled balloon, which rises straight up into the air. (It is a windless da.) fter 4 seconds, Davis uses his clinometer to site the angle of elevation to the balloon at 3. If Davis ee height is 4 feet, 6 inches, what is the height of the balloon after 4 seconds? 40 Geometr Connections Parent Guide
nswers. In each case the slope ratio is 4 = 4. 2. he slope ratio is = 4 4 = 3 3 =. 3. he slope ratio is 3. 4. he slope ratio is 4.. tan28 = z 4 3.2,!z! 7.44 6. tan 70 =,!m! 2.7,!" = 20 m 7. tan 33 =,!! 36.38,!" = 7 8. c! 9.67 (Pthagorean heorem) 2 9.! = 4,! = 2.2. tan = w,!w! 2.9 47. tan 7 = h ; the ladder reaches about 8.66 feet up the wall. 7 ft 2. tan 3 = h,!!h! 2+ 4.! 2. ; at four 30 seconds the balloon is about 2. feet above the ground. 3 4. ft Chapter 4: rigonometr and Probabilit 4 30 ft
PROBBILIY 4.2. - 4.2. lthough the definition of probabilit is simple, calculating a particular probabilit can be trick at times. When calculating the probabilit of flipping a coin and having it come up tails, we can easil see that there are onl two possibilities and one successful outcome. But what if neither the total number of outcomes nor the total number of successes is obvious? In this case, we need to have an accurate wa to count the number of these events. In these sections of the tet, we look at three models to do this: making a sstematic list, making a tree diagram, and making an area model. Each different model has its strengths and weakness, and is appropriate in different situations. See the Math Notes bo on page 29. Eample s Ms. Dobb prepares the week s dinner menu for the students, she has certain rules that she must follow. She must have a meat dish and a vegetable at each dinner. She has four choices for her entree: chicken, fish, beef, and pork. er list of choices for vegetables is a bit larger: peas, carrots, broccoli, corn, potatoes, and turnips. Considering just the meat and the vegetable, what is the probabilit that the first meal she makes will have meat and a green vegetable? o determine the probabilit of a meal of meat and a green vegetable, we need to know how man different meals are possible. hen we need to count how man of the meals have meat and a green vegetable. o count all of the possible meals, we will make a sstematic list, pairing each entree with a vegetable in an organized wa. Chicken Fish Beef Pork Chicken and peas Fish and peas Beef and peas Pork and peas Chicken and carrots Fish and carrots Beef and carrots Pork and carrots Chicken and broccoli Fish and broccoli Beef and broccoli Pork and broccoli Chicken and corn Fish and corn Beef and corn Pork and corn Chicken and potatoes Fish and potatoes Beef and potatoes Pork and potatoes Chicken and turnips Fish and turnips Beef and turnips Pork and turnips From this list we can count the total number of meals: 24 meals. hen we count the number of meals with meat and a green vegetable (peas or broccoli). here are si such meals. herefore the probabilit of the first meal having meat and a green vegetable is 6 24 = 4. 42 Geometr Connections Parent Guide
Eample 2 What is the probabilit of flipping a fair coin 4 times and have tails come up eactl two of those times? o solve this problem, we could make a sstematic list as we did with the previous eample, but there is another technique that works well with this tpe of problem. Since each flip gives us onl two outcomes, we can organize this information in a tree diagram. he first flip has onl two possibilities: heads () or tails (). From each branch, we split again into or. We do this for each flip of the coin. he final number of branches at the end tells us the total number of outcomes. In this problem, there are 6 outcomes. We now count the number of paths along the branches that have eactl two s. One path consisting of is highlighted. he others are,,,, and, for a total of si paths. hus the probabilit of flipping a st flip 2 nd flip 3 rd flip 4 th flip coin four times and having come up eactl two times is 6 6 8 Eample 3 Romeo the rat is going to run through a maze to find a block of cheese. he floor plan of the maze is shown at right, with the cheese to be placed in either section or B. If ever time Romeo comes to a split in the maze he is equall like to choose an path in front of him, what is the probabilit he ends up in section? B o answer this question we will construct an area model to represent this situation. Using an area model is like turning the problem into a dartboard problem. It is eas to see what the probabilit of hitting the shaded portion on the dartboard at right is because the shaded portion makes up one-fourth of the board. herefore the probabilit of hitting the shaded portion is 4. What we want to do is turn the maze problem into a dartboard with the outcome we want (our success) represented b the shaded part. o begin, we start with a square dartboard. You can think of this as being a square. When Romeo comes to the first branch in the maze, he has two choices: a top path and a bottom path. We represent this on the dartboard b splitting the board into two same sized (equall likel) pieces. hen consider Chapter 4: rigonometr and Probabilit 43
what happens if Romeo chooses the bottom path first. If he chooses the bottom path, he comes to another split with two choices, each equall likel. On the area model (dartboard) we show this b splitting the bottom rectangle into two equall like sections. B With one branch, Romeo will end up in section ; with the other branch he will end up in B. We indicate this b putting the letters in the regions representing these outcomes. Note: ou can split the bottom rectangle in half with a top rectangle and a bottom rectangle as well. Since we are ultimatel going to consider the area covered with an, it can be split in an wa as long as the pieces are equal in size. Now consider the top path. If Romeo takes the top path at the first split, he quickl comes to another split where again he has a choice of a top path or a bottom path. Once again we split the top rectangle into two same-sized rectangles since each path is equall likel. One bo will represent the top path and one will represent the bottom. If Romeo takes the lower path, he will end up in section. We indicate this b choosing one of the new regions as representing the lower path, and writing an in that portion. If Romeo takes the upper path, he comes to another split, each equall likel. his means the last section of the dartboard that is not filled in needs to be cut into two equal parts, since each path is equall likel. One of the paths will lead directl to section, the other to section B. Now we can fill in those letters as well. B B B B looking at the dart board now, we can see that since takes up more of the board, we would be more likel to hit section. But to find the actual probabilit, we must determine how much area the sections marked with take up. Recall that this is a! square. We can find the fraction of the area of each part. Remember: the ke is that we divided regions up into equal parts. he lengths of each side of each rectangle is shown on the eterior of the square, while the area is written within the region. We want to know the probabilit of getting into section, which is represented b the shaded portion of the dartboard. he area of the shaded region is 2 4 4 2 4 8 8 = 4 + 4 + 8 2 4 4 = 2 8 + 2 8 + 8 = 8 herefore the probabilit of Romeo wandering into section is 8. his means the probabilit that he wanders into section B is 3 8 since the sum of both probabilities must be. 44 Geometr Connections Parent Guide
Eample 4 In the previous problem, if we let Romeo run through the maze randoml 80 times, how man times would ou epect him to end up in section? In section B? Now that we know the probabilit of Romeo wandering into each of the sections, we can figure out how man times we would epect him to reach each section. Since the probabilit of Romeo wandering into section is 8, we would epect Romeo to end up in section 0 out of 80 times. Similarl, we would epect Romeo to wander into section B 30 times out of 80. his does not mean that Romeo will definitel wander into 0 times out of 80. We are dealing with probabilities, not certainties, and this just gives us an idea of what to epect. Problems. If Keisha has four favorite shirts (one blue, one green, one red, and one ellow) and two favorite pairs of pants (one black and one brown), how man different favorite outfits does she have? What is the best wa to count this? 2. Each morning aron starts his da with either orange juice or apple juice followed b cereal, toast or scrambled eggs. ow man different morning meals are possible for aron? 3. Eliza likes to make dail events into games of chance. For instance, before she went to bu ice vanilla utti fruitti cream at the local ice cream parlor, she created two spinners. he first has her three favorite chocolate flavors while the second has Cone and Dish. Eliza will order whatever comes up on the spinners. What is the probabilit that she will be eating tutti fruitti ice cream from a dish? cone dish 4. Bart is going to flip a coin three times. What is the probabilit that he will see at least two tails?. Mr. Fudge is going to roll two fair dice, one green and one red. What is the probabilit that the sum will be four or less? $.00 6. Welcome to another new game show, $.00 00 Spinning for Luck! s a contestant, ou will $0 $00 be spinning two wheels. he first wheel determines a possible dollar amount that ou $.00 $2.00-2 could win. he second wheel is the multiplier. You will multipl the two results of our spin to determine the amount ou will win. Unfortunatel, ou could owe mone if Dollar amount he Multiplier our multiplier lands on -2! What is the probabilit that ou could win $0 or more? What is the probabilit that ou could owe $0 or more? Chapter 4: rigonometr and Probabilit 4
For problems 7-, a bag contains the figures shown below right. If ou reach in and pull out a shape at random, what is the probabilit that ou pull out: 7. a figure with at least one right angle? 60 8. a figure with an acute angle? 9. a shape with at least one pair of parallel sides? 70. a triangle? nswers. Eight different outfits. sstematic list works best. 2. Si meals. sstematic list or tree diagram works. 3. 3! 2 = 6 4. 2 (See the tree diagram in eample 2.). 36 = 8 6. Winning $0 or more: 2, owing $0 or more: 2 2 7. 8. 3 9. 2. 2 46 Geometr Connections Parent Guide