Think Unit bout 6 This Lesson Situation 1 Investigation 1 Name: Think about the design and function of this automobile jack. Use the uto Jack custom tool to test ou ideas. a bout how long would the theaded od need to be if the jack is to be stoed with points B and D as close togethe as possible? b s the distance C deceases, how do the angle measues of BC change? How does the distance between point B and the theaded od change? c How does the height of the jack, BD, compae to the length of the altitude of BC dawn fom point B? Eplain ou thinking. d Suppose the jack is set so that C is as long as possible. s the theaded od is tuned at a constant ate, the distance C deceases at a constant ate. How would ou descibe the ate at which the height BD of the jack changes? In this lesson, ou will use angles in a coodinate plane to define tigonometic functions and use those functions to detemine measues of unknown pats in ight tiangles. These functions also fom the basis fo finding unknown pats of an tiangle, as ou will lean in the second lesson. Thoughout this unit, ou will see the pactical powe of tigonometic methods fo solving a wide ange of applied poblems. Investigation 1 Connecting ngle Measues and Linea Measues One of the most effective stategies fo calculating distances that cannot be measued diectl is to epesent the situation with a tiangle in which the length of one side is the desied distance and othe sides and/o angles can be measued. Tigonomet povides methods fo using the known pats of a tiangle to calculate those that ae unknown. The ke connections ae the tigonometic atios sine, cosine, and tangent defined as functions of angles in standad position in a coodinate plane. s ou wok on the poblems in this investigation, look fo answes to the following questions: How ae the sine, cosine, and tangent functions defined? How can thei values be estimated? LESSON 1 Tigonometic Functions 459
1 Stud the diagam below of a adio tansmitte towe with two suppot wies attached to it at and E and to the gound at B and D. Fist, focus on the tiangle fomed b the towe, the gound, and the shote suppot wie. E B C D a. If CD = 50 feet and CE = 100 feet, is the size and shape of ight DCE completel detemined? Wh o wh not? b. How long is suppot wie DE? c. If ou wee asked to attach a suppot wie, 125 feet long, fom point D to the towe, how fa up the towe would ou have to climb? d. What elationship among sides of a ight tiangle have ou used in Pats b and c? 2 Now focus on the ight tiangle fomed b the towe, BC, and the longe suppot wie B. a. If BC = 75 feet and m BC = 66, is the size and shape of ight BC completel detemined? Wh o wh not? b. Eplain wh the size and shape of a ight tiangle ae completel detemined when the measues of two sides ae known o the measue of one acute angle and the length of one side ae known. c. It should be possible to use the infomation given in Pat a to find C, the distance above the gound at which suppot wie B is attached to the towe. Will the Pthagoean Theoem help? Eplain. 460 UNIT 7 Tigonometic Methods
To solve poblems like Poblem 2 Pat c, ou need to find a connection between angle measues and segment lengths in a ight tiangle. In this case, it is helpful to think of an angle as being fomed b otating a a about its endpoint fom an initial position to a teminal position. teminal side 66 B initial side C The point about which the a is otated is the vete of the angle. The initial position of the a is the initial side of the angle. The teminal position of the a is the teminal side of the angle. To indicate the diection of otation fom initial side to teminal side, it is customa to sa the angle has positive measue if the otation is counteclockwise and has negative measue if the otation is clockwise. 3 The diagam at the ight shows fou points on the teminal side of an angle in standad position in a V(12, 9) coodinate plane. The vete of U(8, 6) the angle is at the oigin and its W(2, 1.5) Z(4, 3) initial side coincides with the positive -ais. a. Fo each point (, ) shown on the teminal side, find the atios _. O i. U(8, 6) ii. V(12, 9) iii. W(2, 1.5) iv. Z(4, 3) b. How do the atios _ compae in each case? Wh does that make sense? c. Fo each point (, ) in Pat a, suppose is the distance fom the oigin to the point. How do ou think the atios _ would compae in each case? The atios _? Check ou conjectues. 4 You discoveies about the atios of lengths in Poblem 3 appl to an angle in standad position. Conside the diagam below that shows an angle with degee measue θ (Geek lette theta ) in standad position. P (c, d) P(a, b) O θ Q Q LESSON 1 Tigonometic Functions 461
Let P(a, b) and P'(c, d) be an two points on the teminal side O, othe than the oigin O. a. Find the slope of O using points O and P. Find the slope of O using points O and P'. b. How ae _ PQ and _ P'Q' elated? Wh? OQ OQ' c. Eplain wh OP'Q' is the image of OPQ unde a size tansfomation with cente at the oigin and magnitude k = _ c a. Use the following questions to guide ou thinking. i. What is the image of point O unde this tansfomation? ii. Wh is point Q' the image of point Q unde this tansfomation? iii. Wh is point P' the image of point P unde this tansfomation? d. Use ou wok in Pat c to help eplain each step in the easoning below. i. OQ' = k(oq) and OP' = k(op). ii. OP' = k(op) and P'Q' = k(pq). So, _ OQ' OQ = _ OP' OP. OP' So, _ OP = _ P'Q' PQ. So, OQ' OP = OQ OP'. So, OP' PQ = OP P'Q'. So, _ OQ OP = _ OQ' OP'. So, _ PQ OP = _ P'Q' OP'. e. Wite a statement descibing how ou wok in Pats b and d suppot ou discoveies in Poblem 3. f. Daw diagams simila to that above fo cases whee θ > 90. Does easoning simila to that in Pats b d hold in these cases? Eplain. You wok in Poblem 4 shows that if P(, ) is a point (not the oigin) on the teminal side of an angle in standad position and = 2 + 2, then the atios _, _, and _ do not depend on the choice of P; the depend onl on the measue of POQ. That is, these atios ae functions of the measue θ of the angle. These functions ae called tigonometic functions and ae given special names as indicated below. tangent of θ = tan θ = _ ( 0) sine of θ = sin θ = _ = 2 + 2 P(, ) cosine of θ = cos θ = _ O θ Q 5 The teminal side of an angle in standad position with measue θ contains the given point. In each case, daw the angle on a coodinate gid. Then find cos θ, sin θ, and tan θ. a. P(12, 5) b. P(-6, 4) c. Fo an angle with measue θ, is it possible fo sin θ > 1? Fo cos θ > 1? Fo tan θ > 1? 462 UNIT 7 Tigonometic Methods
6 The diagam below shows a potion of a cicle with adius 10 cm, dawn on a 2-mm gid. ngles ae maked off in 10 intevals, so m OP 1 = 10, m OP 2 = 20, m OP 3 = 30, and so on. You can use this diagam to calculate appoimate values of cos θ, sin θ, and tan θ fo angles with measue θ between 0 and 90. 11 10 P 9 P 8 P 7 9 P 6 8 P 5 7 P 4 (7.7, 6.4) 6 5 P 3 4 P 2 3 2 P 1 1 P 0 O 1 2 3 4 5 6 7 8 9 10 11 a. Veif the enties in the table below fo the case of OP 4. b. Make a cop of the table and then use the diagam to find the missing enties. Shae the wokload. P n m OP n cos θ sin θ tan θ 0 20 9.4 3.4 40 7.7 6.4 0.77 0.64 0.84 60 0.50 80 90 LESSON 1 Tigonometic Functions 463
7 In Poblem 6, ou wee able to find appoimate values of the tigonometic functions of 20, 40, 60, and 80. In the case of angles with measue 0 and 90, ou wee able to detemine eact values (with the eception of tan 90 ). You can use geometic easoning to find eact values of tangent, sine, and cosine of 45. a. Daw an angle of 45 in standad position. b. What is an equation fo the line that makes an angle of 45 with the positive -ais? c. What do ou know about the coodinates of an point on the teminal side of this angle? d. Choose a point (, ) on the teminal side of the angle and then find tan 45, sin 45, and cos 45. Give eact values, not decimal appoimations. 8 Now etun to Poblem 2 about the adio tansmitte towe. Suppose BC = 75 feet and m BC = 66. (75, ) E 66 B C D B C(75, 0) To detemine C, the distance above the gound at which suppot wie B is attached to the towe: a. Daw BC in standad position as in the diagam above on the ight. Eplain wh the coodinates of points and C ae labeled as shown. b. Wite an epession fo tan 66 in tems of the given infomation. c. Use the diagam in Poblem 6 to calculate an appoimate value of tan 66. d. Using ou esults fom Pats b and c, find the appoimate height C. e. How long is suppot wie B? f. Show how ou could use a tigonometic function to find the appoimate length of B without fist finding the height C. Compae ou answe to that obtained in Pat e. 464 UNIT 7 Tigonometic Methods
9 The values of the tigonometic functions ou found b using the gid on page 463 wee ough appoimations of the function values. Seveal centuies ago, mathematicians spent eas calculating values of the tigonometic functions b hand to seveal decimal places so the could be used in suveing and astonom. Toda, moe advanced mathematical methods ae used. You will stud those methods in a late couse. potion of a table of tigonometic function values with fou-digit accuac is shown below. ngle sin cos tan ngle sin cos tan 45 0.7071 0.7071 1.0000 61 0.8746 0.4848 1.8040 46 0.7193 0.6947 1.0355 62 0.8829 0.4695 1.8807 47 0.7314 0.6820 1.0724 63 0.8910 0.4540 1.9626 48 0.7431 0.6691 1.1106 64 0.8988 0.4384 2.0503 49 0.7547 0.6561 1.1504 65 0.9063 0.4226 2.1445 50 0.7660 0.6428 1.1918 66 0.9135 0.4067 2.2460 51 0.7771 0.6293 1.2349 67 0.9205 0.3907 2.3559 52 0.7880 0.6157 1.2799 68 0.9272 0.3746 2.4751 53 0.7986 0.6018 1.3270 69 0.9336 0.3584 2.6051 54 0.8090 0.5878 1.3764 70 0.9397 0.3420 2.7475 55 0.8192 0.5736 1.4281 71 0.9455 0.3256 2.9042 56 0.8290 0.5592 1.4826 72 0.9511 0.3090 3.0777 57 0.8387 0.5446 1.5399 73 0.9563 0.2924 3.2709 58 0.8480 0.5299 1.6003 74 0.9613 0.2756 3.4874 59 0.8572 0.5150 1.6643 75 0.9659 0.2588 3.7321 60 0.8660 0.5000 1.7321 a. Compae the values of sine, cosine, and tangent of 45 that ou found in Poblem 7 with the values in the table above. b. Compae the values of tan 66 and cos 66 that ou used in Poblem 8 with the values in the table above. c. s the measue of an angle inceases fom 45 to 75, i. how does the sine of the angle change? ii. how does the cosine of the angle change? iii. how does the tangent of the angle change? d. Wh do the pattens of change in Pat c make sense in tems of the diagam on page 463? LESSON 1 Tigonometic Functions 465
e. Use the table to detemine each of these tigonometic function values. i. sin 58 ii. cos 48 iii. tan 72 f. Given the function values below, use the table to detemine the measue of the angle θ whose teminal side is in the fist quadant. i. sin θ = 0.8910 ii. cos θ = 0.5878 iii. tan θ = 1.1106 Summaize the Mathematics In this investigation, ou eploed the sine, cosine, and tangent, thee membes of a new famil of functions called tigonometic functions. a How do each of these functions povide a connection between angle measue and linea measue? b Suppose the teminal side of an angle in standad position with measue θ lies on the line with equation = 2, 0. i. Find tan θ. ii. Find cos θ. iii. Find sin θ. c Suppose θ is the measue of an angle in standad position whose teminal side is in the fist quadant and tan θ = 4 _ 5. i. Find cos θ. ii. Find sin θ. d Descibe the patten of change fo each function as the measue of an angle in standad position inceases fom 0 to 90. i. cos θ ii. sin θ iii. tan θ Be pepaed to eplain ou esponses to the class. Check You Undestanding The teminal side of an angle in standad position with measue θ contains the point P(4, 7). a. Daw a sketch of the angle. b. Find sin θ, cos θ, and tan θ. c. Use the table on page 465 to estimate θ to the neaest degee. 466 UNIT 7 Tigonometic Methods