INVESTIGATION 2 Wht s the Angle? In the previous investigtion, you lerned tht when the rigidity property of tringles is comined with the ility to djust the length of side, the opportunities for useful ppliction expnd gretly. You proly noticed tht the methods used to determine lengths nd ngle mesures involved mesuring the models you mde. In this investigtion, you will use right tringles nd similrity to explore other wys in which lengths nd ngle mesures cn e determined. Recll tht if two figures re similr with scle fctor of s, then corresponding ngles re congruent ( ) nd lengths of the corresponding sides re relted y the multiplier s. The two school crossing signs shown elow re similr. Here pentgon ADE is similr to A D E (ADE ~ A D E ) with scle fctor of 2. corresponds to, so. ( is congruent to.) Segment ED corresponds to segment E D, so 2 ED = E D or, equivlently, ED = E D. 2 1. Imgine tht you nd clssmte ech drw tringle with three ngles of one tringle congruent to three ngles of the other tringle. Do you think the two tringles will e similr? Mke conjecture.. Now conduct the following experiment. Hve ech memer of your group drw segment (no two with the sme length). Use protrctor to drw 50 ngle t one end of the segment. Then drw 60 ngle t the other end of the segment to form tringle. Wht should e the mesure of the third ngle? heck your nswer. Are these tringles similr to one nother? Wht evidence cn you give to support your view?. Repet Prt with ngles mesuring 40 nd 120. Are these tringles similr? Give evidence to support your clim. LESSON 2 TRIANGLES AND TRIGONOMETRI RATIOS 395
c. Which, if ny, of the following sttements do you think re lwys true? Justify your response with resons or counterexmple. If one tringle hs three ngles congruent to three corresponding ngles on nother tringle, then the tringles re similr. If one tringle hs two ngles congruent to two corresponding ngles on nother tringle, then the tringles re similr. If one tringle hs one ngle congruent to one ngle on nother tringle, then the tringles re similr. 2. Now pply your discoveries in Activity 1 to the specil cse of right tringles.. Ech group memer should drw segment A (ech different length). Using your segment A s side, drw A with A mesuring 35 nd right ngle t. It is importnt to drw your tringle very crefully. Wht is the mesure of the other ngle ( )?. Are the tringles your group memers drew similr? Explin. c. hoose the smllest tringle drwn in Prt. Determine the pproximte scle fctors relting this tringle to the others drwn y group memers. For right tringle A, it is stndrd procedure to lel the right ngle with the cpitl letter hypotenuse side opposite A or nd to lel the sides opposite c side djcent to the three ngles lower cse,, 35 nd c s shown. omplete A leling of your tringle in this wy. (Additionl wys to refer side opposite or to the sides of right tringle re side djcent to A lso included in the digrm. The hypotenuse is lwys the side opposite the right ngle, ut the designtion of the other sides depends on which ngle is considered.) 3. A digrm of right A is given elow. Give the mesures of the following ngles or sides: A. 64. 20 cm c. Side opposite A 8.8 cm d. Side (leg) djcent to e. Side (leg) djcent to A 18 cm 396 UNIT 6 GEOMETRI FORM AND ITS FUNTION
4. Refer to the right tringles your group drew for Activity 2.. Mke tle like the one elow. Ech group memer should choose unit of mesure. Then crefully mesure nd clculte the indicted rtios for the right A tht you drew. Express the rtios to the nerest 0.01. Investigte ptterns in the three rtios for your group s tringles. Right Tringle Side Rtios Rtio Student 1 Student 2 Student 3 Student 4 c c. ompre the rtios from your group with those of other groups. c. On the sis of Prts nd, mke conjecture out the three rtios in the tle for ny right A with 35 ngle t A. d. How could the three rtios e descried in terms of the hypotenuse nd the sides opposite nd djcent to A? In terms of the hypotenuse nd the sides opposite nd djcent to? e. Mke conjecture out the three rtios in the tle for ny right tringle with 35 ngle. f. Mke conjecture out the three rtios for ny right tringle with 55 ngle. 5. As group, drw severl exmples of right A in which A hs mesure of 40 nd hs mesure of 90.. ompute the three rtios c, c, nd. Record the rtios in tle like the one in Activity 4.. Wht pttern do you see in these rtios? c. How is the pttern for these rtios similr to the pttern for the rtios in Activity 4? How is it different? d. Wht seems to cuse the differences in the results from the two ctivities? Test your conjecture y experimenting with nother set of similr right tringles nd inspecting the rtios. LESSON 2 TRIANGLES AND TRIGONOMETRI RATIOS 397
You hve oserved tht s the mesure of n cute ngle chnges in right tringle, rtios of the lengths of the sides lso chnge. In fct, ech rtio is function of the size of the ngle. These reltionships re importnt ecuse they relte mesures of ngles (in degrees) to rtios of liner mesures (in centimeters, miles, nd so on). The reltionships or functions hve specil nmes. For right tringle A with sides,, nd c, the sine, cosine, nd tngent rtios for A re defined s follows: c A sine of A = sin A = cosine of A = cos A = tngent of A = tn A = length of side opposite A length of hypotenuse These rtios re clled trigonometric rtios. Sine, cosine, nd tngent re similrly defined y forming the rtios using the sides opposite nd djcent to. The revitions re sin, cos, nd tn. 6. Refer to the tringles you drew for Activity 5. Write the definitions for sin, cos, nd tn, nd then find sin 50, cos 50, nd tn 50. 7. Suppose you hve right tringle with n cute ngle of 27.5. One wy you could find the sine, cosine, nd tngent of 27.5 would e to mke very ccurte right tringle nd mesure. Another wy is to use your clcultor.. To clculte trigonometric rtio for n ngle mesured in degrees, first e sure your clcultor is set in degree mode. Then simply press the keys corresponding to the rtio desired. For exmple, to clculte sin 27.5 on most grphing clcultors, press SIN 27.5 ENTER. Try it. Then clculte cos 27.5 nd tn 27.5.. ompre sin 27.4 nd sin 27.6 to your vlue for sin 27.5. How mny deciml plces should you include to show tht the ngle whose sine you re finding ws mesured to the nerest 0.1? c. How mny deciml plces should you report for sin 66.5 to indicte tht the ngle ws mesured to the nerest 0.1? d. Use your clcultor to find the sine, cosine, nd tngent of 35 nd of 50. ompre these results with those you otined y mesuring in Activities 4 through 6. = c length of side djcent to A length of hypotenuse length of side opposite A length of side djcent to A = c = 398 UNIT 6 GEOMETRI FORM AND ITS FUNTION
You cn use your clcultor to compute vlues for sine, cosine, nd tngent of ny ngle. Severl hundred yers go mthemticins spent yers clculting these rtios y hnd to severl deciml plces so tht they could e used in surveying nd stronomy. Until recently, efore scientific nd grphing clcultors ecme common, people usully looked up the rtio vlues from lrge tle. Now tht clcultor replces this tedious work, you cn concentrte on understnding trigonometric rtios nd their uses. Knowing some informtion out tringle or pir of tringles often llows you to conclude other informtion. c d heckpoint Wht cn you conclude out two tringles if two ngles of one re congruent to two ngles of the other? Why does knowing the mesure of n cute ngle of right tringle completely determine the tringle s shpe? For two different tringles A nd DEF, in which A nd D oth hve mesure x nd nd F re right ngles, wht cn you sy A F out the rtios A? Explin why this mkes sense. nd D DE If sin x = c in right A, then which ngle hs mesure x? Which ngle hs mesure x when cos x = c? e prepred to discuss your responses with the entire clss. On Your Own Refer to the drwing of PQR elow.. Which is the side opposite P? The side djcent to R? The hypotenuse?. Use the Pythgoren Theorem to find the length of side PQ. c. Find the following trigonometric rtios: sin P sin R tn P cos R P d. Vlerie mesured R nd found it to e lmost 74. Use clcultor nd your results from Prt c to estimte the mesure of R to the nerest 0.1. 25 ft R 7 ft Q LESSON 2 TRIANGLES AND TRIGONOMETRI RATIOS 399
INVESTIGATION 3 Mesuring Without Mesuring Shown elow is hicgo s t olumn, sculpture y les Oldenurg. 1. In your group, rinstorm out possile wys to determine the height of the sculpture.. hoose one method nd write detiled pln.. Trde plns with nother group nd compre the two plns. c. Wht ssumptions did the other group mke in devising its pln? d. Which pln seems esier to crry out? Why? Your clss proly thought of severl plns to determine the height of t olumn. For exmple, one could use n extension ldder on fire truck to clim to the top nd drop weighted nd mesured cord to the ground. This would e direct mesurement procedure. 2. An indirect wy to mesure the height of t olumn would e to use right tringle nd trigonometric rtio.. In the sitution depicted elow, wht lengths nd ngles could you determine esily y direct mesurement (nd without using high-powered equipment)? A. Which trigonometric rtios of A involve side? Of these, which lso involve mesurle length? 400 UNIT 6 GEOMETRI FORM AND ITS FUNTION
c. Which of the trigonometric rtios of involve side nd mesurle length? If you know the size of A, how cn you find the mesure of? d. Krist nd D wn decided to find the height of t olumn themselves. First Krist chose spot to e point A, 20 meters from the sculpture (point ). D wn used clinometer, like the one shown t the right, to estimte the mesure of A (the ngle of elevtion from the horizontl to the top of the t). He mesured A to e 55. Wht is the mesure of? e. Krist nd D wn proceeded to find the height of the t independently s shown elow. D'wn I need to find so tht A = tn55. ut tn 55 = 1.43 nd A = 20 m. So I need to solve 20 = 1.43. If I multiply the eqution y 20, I get = 1.43 20 = 28.6 m tn35 = A 0.7 = 20 Multiplying the eqution y, I get 0.7 = 20. Dividing y 0.7, I get = 20 or 28.6 m. 0.7 Krist Anlyze D wn s thinking. Why did he multiply y 20? Anlyze Krist s thinking. Why did she multiply y? Why did she divide y 0.7? Are the nswers correct? Explin your response. f. How could you use Krist s nd D wn s work to help estimte the height of t olumn? LESSON 2 TRIANGLES AND TRIGONOMETRI RATIOS 401
g. Kim sid he could find the length A (the line of sight distnce) y solving cos 55 = A. Anlyze A Kim s thinking shown here. Explin Kim s thinking. Is Kim correct? Wht is nother wy Kim could hve found A using trigonometric rtios? ould you find A without using trigonometric rtios? Explin your resoning. 3. Ech prt elow gives dt for right A. Sketch model tringle nd then, using your clcultor, find the lengths of the remining two sides.. = 52, = 5 m. A = 78, = 5 mi c. A = 21, = 8 in. d. = 8, = 8 ft e. = 37, c = 42 yd f. A = 82, c = 14 cm 4. Terri is flying kite nd hs let out 500 feet of string. Her end of the string is 3 feet off the ground.. If KIT hs mesure of 40, pproximtely how high off the ground is the kite?. As the wind picks up, Terri is le to fly the kite t 56 ngle with the horizontl. Approximtely how high is the kite? cos55 = 20 A This is equivlent to A = 20 cos 55 A = 20 0.57 A = 34.9 Kim c. Wht is the highest Terri could fly the kite on 500 feet of string? Wht would e the mesure of T KIT then? I E d. Experiment with your clcultor to estimte the mesure of KIT needed to fly the kite t height of 425 feet. In the previous situtions, you used trigonometric rtios to determine n unknown or inccessile distnce. In Activity 4 Prt d, you proly found wy to find the mesure of n ngle when you know the lengths of two sides in right tringle. K 402 UNIT 6 GEOMETRI FORM AND ITS FUNTION
5. Estimte (to the nerest degree) the mesure of cute ngle for ech of the following trigonometric rtios of. heck your estimte in ech cse y drwing model right A, using sides whose lengths give the pproprite rtio, nd then mesuring.. sin = 3 5. cos = 1 2 c. tn = 4 5 d. onsider how you found the mesure of n cute ngle when you knew trigonometric rtio for right tringle with tht ngle mesure. ompre your group s pproch with the pproches of other groups. 6. You know how to use clcultor to produce trigonometric rtio when you know the mesure of n ngle. You lso cn use clcultor to produce the ngle when you know trigonometric rtio s in Activity 5.. Suppose you know sin A = 4 = 0.8. Use the 5 sin 1 function of your clcultor to compute the ngle whose sine is 0.8. (Mke certin your clcultor is set in degree mode.). Wht would you get if you clculted the sine of the ngle in the clcultor disply t the right? c. Use your clcultor to find the mesure of tht corresponds to ech of the rtios given in Activity 5. ompre these vlues to the vlues you otined in tht ctivity. d. Use your clcultor to find the mesure of the ngle in ech of the following cses. tn = 1.84 sin A = 0.852 cos = 0.213 7. The ndin Ntionl Tower in Toronto, Ontrio, is pproximtely 553 meters tll. This tower is the tllest free-stnding structure in the world.. Sketch the tower nd dd the fetures descried in Prts, c, nd d s you work to nswer ech prt.. At some time on sunny dy, the sun mkes the tower cst 258-meter shdow. Wht is the mesure of the ngle formed y sun ry nd the ground t the tip of the shdow? 1 sin (.8) 53.13010235 LESSON 2 TRIANGLES AND TRIGONOMETRI RATIOS 403
c. From the top of the ndin Ntionl Tower, ot is oserved in Lke Ontrio, pproximtely 8,000 meters wy from the se of the tower. Assume the se of the tower is pproximtely level with the lke surfce. Wht ngle elow the horizontl must the oserver look to see the ot? d. Estimte the line of sight distnce from the oserver to the ot in Prt c. Find this distnce using trigonometric rtios nd without using them. 8. Lkeshi is out 1.7 meters tll. When stnding 5 meters from her school uilding, her ngle of sight to the top of the uilding is 75.. Estimte the height of the uilding.. Suppose Lkeshi moves to position 10 meters from the uilding. Wht is the ngle of her new line of sight to the top of the uilding? c. Mrcio, who is lso out 1.7 meters tll, is stnding on top of the uilding. He sees Lkeshi stnding 15 meters from the uilding. At wht ngle elow the horizontl is his line of sight to Lkeshi? heckpoint Trigonometric rtios re useful to clculte lengths nd ngles in righttringle models. Refer to the right tringle shown elow in summrizing your thinking out how to use trigonometric rtios. A c c If you knew nd the mesure of, how would you find? Wht clcultor keystroke sequence would you use? If you knew nd c, how would you find the mesure of A? Wht clcultor keystroke sequence would you use? If you knew nd the mesure of, how would you find c? Wht clcultor keystroke sequence would you use? e prepred to explin your methods to the whole clss. 404 UNIT 6 GEOMETRI FORM AND ITS FUNTION
The ngle of elevtion to the top of n oject is the ngle formed y the horizontl nd the line of sight to the top of the oject. In the digrm elow, AD is the ngle of elevtion. The ngle of depression to n oject is the ngle formed y the horizontl nd the line of sight to the oject elow. In the digrm, A is the ngle of depression. Angle of depression A Line of sight Angle of elevtion D On Your Own A person on n oil-drilling ship in the Gulf of Mexico sees semi-sumersile pltform with tower on top of it. The tower stnds 130 meters ove the pltform floor. 21 Pltform. If the oserver s position on the ot is 15 meters under the floor of the pltform nd the ngle of elevtion to the top of the rig is 21, wht three distnces cn you find? Find them.. Suppose the ot moves so tht it is 200 meters from the center of the oil rig. Wht is the ngle of elevtion now? LESSON 2 TRIANGLES AND TRIGONOMETRI RATIOS 405