ontents UNIT 7 TRIGONOMETRI METHODS Lesson 1 Trigonometric Functions................... 462 1 onnecting ngle Mesures nd Liner Mesures.............. 463 2 Mesuring Without Mesuring......................... 471 3 Wht s the ngle?............................... 475 On Your Own................................. 478 Lesson 2 Using Trigonometry in ny Tringle............ 492 1 The Lw of Sines............................... 493 2 The Lw of osines.............................. 497 3 Tringle Models Two, One, or None?..................... 502 On Your Own................................. 507 Lesson 3 Looking ck............................ 520 UNIT 8 PROILITY DISTRIUTIONS Lesson 1 Proility Models....................... 522 1 The Multipliction Rule for Independent Events................ 523 2 onditionl Proility............................ 528 3 The Multipliction Rule When Events re Not Independent.......... 532 On Your Own................................. 536 Lesson 2 Expected Vlue......................... 545 1 Wht s Fir Price?.............................. 546 2 Expected Vlue of Proility Distriution................. 549 On Your Own................................. 552 Lesson 3 The Witing-Time Distriution............... 560 1 Witing for Doules.............................. 561 2 The Witing-Time Formul........................... 565 3 Expected Witing Time............................ 570 On Your Own................................. 573 Lesson 4 Looking ck.......................... 586 Glossry........................................ 590 Index of Mthemticl Topics............................ 607 Index of ontexts................................... 616 xi
UNIT 7 TRIGONOMETRI Trigonometry, or tringle mesure, is n importnt tool used y surveyors, nvigtors, engineers, uilders, stronomers, nd other scientists. Tringultion nd trigonometry provide methods to indirectly determine otherwise inccessile distnces nd ngle mesures. The sme tools re useful in the design nd nlysis of mechnisms involving tringles in which the lengths of two sides of tringle re fixed while the length of the third side is llowed to vry. Through your work on the investigtions in this unit, you will develop the understnding nd skill needed to solve prolems using trigonometric methods. Key ides will e developed in two lessons. METHODS Lessons 1 Trigonometric Functions Use ngles in stndrd position in coordinte plne to define the trigonometric functions sine, cosine, nd tngent. Interpret nd pply those functions in the cse of situtions modeled with right tringles. 2 Using Trigonometry in ny Tringle Develop the Lw of Sines nd the Lw of osines, nd use those reltionships to find mesures of sides nd ngles in tringles. Solve equtions involving severl vriles for one of the vriles in terms of the others.
LESSON 2 Using Trigonometry in ny Tringle In Lesson 1, you lerned severl strtegies for using the Pythgoren Theorem nd the trigonometric functions (sine, cosine, nd tngent) to clculte unknown side lengths nd ngle mesures in situtions represented y right tringles. ut often, prolem situtions re modeled y tringles tht re not right tringles. onsider, for exmple, the prolem of developing n ccurte mp of the floor of the Grnd nyon. 488 UNIT 7
Think out This Sitution Suppose surveyor sights point, the tip of pointed spur deep in the cnyon, from tringultion points nd on the south rim. ws mesured to e 2.68 miles. Using trnsit, m ws found to e 64 ; m ws found to e 34. Drw nd lel tringle representing this sitution. Is the tringle right tringle? How cn you e sure of your nswer? c Wht side lies opposite the 34 ngle in your tringle? Is sin 34 equl to rtio of lengths of two sides of your tringle? If so, which ones? If not, why not? d How might you go out determining the distnces nd? In this lesson, you will investigte two importnt properties of ny tringle tht relte ngle mesures nd side lengths known s the Lw of Sines nd the Lw of osines. These properties will dd to the trigonometric methods ville to you for mking indirect mesurements nd for nlyzing mechnisms in which the lengths of two sides of tringle re fixed ut the length of the third side vries. Investigtion 1 The Lw of Sines If the tringle tht models sitution involving unknown distnces is not (or might not e) right tringle, then it is not so esy to determine the distnces; ut it cn e done. One method tht is sometimes helpful uses the Lw of Sines. s you work on the prolems of this investigtion, look for nswers to the following question: Wht is the Lw of Sines, nd how cn it e used to find side lengths or ngle mesures in tringles? Suppose tht two prk rngers who re in towers 10 miles prt in ntionl forest spot fire tht is fr wy from oth of them. Suppose tht one rnger recognizes the fire loction nd knows it is out 4.9 miles from tht tower. LESSON 2 Using Trigonometry in ny Tringle 489
With this informtion nd the ngles given in the digrm elow, the rngers cn clculte the distnce of the fire from the other tower. Fire 4.9 miles 53 29 10 miles One wy to strt working on this prolem is to divide the otuse tringle into two right tringles s shown elow: h 4.9 miles 29 10 - x D 53 x t first, tht does not seem to help much. Insted of one segment of unknown length, there re now four! On the other hnd, there re now three tringles in which you cn see useful reltionships mong the known sides nd ngles. 1 Use trigonometry or the Pythgoren Theorem to find the length of. When you hve one sequence of clcultions tht gives the desired result, see if you cn find different pproch. 2 In one clss in Settle, Wshington, group of students presented their solution to Prolem 1 nd climed tht it ws the quickest method possile. heck ech step in their resoning nd explin why ech step is or is not correct. (1) _ h = sin 29 (3) _ h = sin 53 4.9 (2) h = sin 29 (4) h = 4.9 sin 53 (5) sin 29 = 4.9 sin 53 4.9 sin 53 (6) = _ sin 29 (7) 8.1 miles ompre your solution from Prolem 1 with this reported solution. 490 UNIT 7 Trigonometric Methods
3 The pproch used in Prolem 2 to clculte the unknown side length of the given tringle illustrtes very useful generl reltionship mong sides nd ngles of ny tringle.. Explin why ech step in the following derivtion is correct for the cute elow. h_ = sin (1) h = sin (2) h_ = sin (3) h = sin (4) sin = sin (5) _ sin = _ sin (6). How would you modify the derivtion in Prt to show tht _ sin = _ sin? c The reltionship derived in Prolem 3 for cute ngles,, nd holds in ny tringle, for ll three of its sides nd their opposite ngles. It is clled the Lw of Sines nd cn e written in two equivlent forms. The cses for right tringle or n otuse tringle re derived in Extensions Tsks 22 nd 23. In ny tringle with sides of lengths,, nd c opposite,, nd, respectively: _ sin = _ sin or equivlently, = _ sin c c h c _ sin = _ sin = c_ sin. You cn use the Lw of Sines to clculte mesures of ngles nd lengths of sides in tringles with even less given informtion thn the fire-spotting prolem t the eginning of this investigtion. In prctice, you only use the equlity of two of the rtios t ny one time. LESSON 2 Using Trigonometry in ny Tringle 491
4 clss in Sn ntonio, Texs, greed on the following representtion of the surveying prolem in the Think out This Sitution (pge 489). Use wht you know out ngles in tringle nd the Lw of Sines to determine the distnces nd. 2.68 miles 64 34 5 Suppose tht two rngers spot forest fire s indicted on the digrm elow. Find the distnces from ech tower to the fire. Fire 135 10 miles 14 Summrize the Mthemtics The Lw of Sines sttes reltion mong sides nd ngles of ny tringle. It cn often e used to find unknown side lengths or ngle mesures from given informtion. Suppose you hve modeled sitution with PQR s shown elow. Wht miniml informtion out the sides nd ngles of PQR will llow you to find the length of QR R using the Lw of Sines? How would you use tht informtion to p clculte QR? q Q Wht miniml informtion out the sides nd ngles of PQR will llow you to find the mesure of Q? How r would you use tht informtion to clculte m Q? P e prepred to explin your thinking nd methods to the entire clss. 492 UNIT 7 Trigonometric Methods