5.3 Investigting nd pplying the Sine Lw in ute Tringles GOL Verify the sine lw nd use it to solve rel-life prolems. INVESTIGTE the Mth The trigonometri rtios sine, osine, nd tngent re defined only for right tringles. In n olique tringle, these rtios no longer pply.? In n olique tringle, wht is the reltionship etween side nd the sine of the ngle opposite tht side? YOU WILL NEED dynmi geometry softwre olique tringle tringle (ute or otuse) tht does not ontin right ngle. Use dynmi geometry softwre to onstrut ny ute tringle.. Lel the verties,, nd. Then nme the sides,, nd s shown.. Mesure ll three interior ngles nd ll three sides. D. Using one side length nd the ngle opposite tht side, hoose lulte... from the Mesure menu to evlute the rtio side length sin(opposite ngle). E. Repet prt D for the other sides nd ngles. Wht do you notie? F. Drg ny vertex of your tringle. Wht hppens to the sides, ngles, nd rtios? Now drg the other two verties nd explin. G. Express your findings in words with mthemtil reltionship Refleting H. re the reiprols of the rtios you found equl? Explin how you know. I. The reltionship tht you verified is the sine lw. Why is this nme pproprite? J. Does the sine lw pply to ll types of tringles (otuse, ute, nd right)? Explin how you know. Teh Support For help using dynmi geometry softwre, see Tehnil ppendix, -2 nd -22. sine lw in ny ute tringle, the rtios of eh side to the sine of its opposite ngle re equl sin 5 sin sin 5 5 sin sin 5 sin or hpter 5 Trigonometry nd ute Tringles 283
PPLY the Mth EXMPLE Using the sine lw to lulte n unknown length Determine x to the nerest tenth of entimetre. 55 70 9.2 m x rr s Solution x sin 55 5 9.2 sin 70 The tringle doesn t hve 90 ngle. So it isn t onvenient to use the primry trigonometri rtios. I hose the sine lw euse I knew two ngles nd the side opposite one of those ngles. I wrote the rtios with the sides in the numertors to mke the lultions esier. sin 55 3 x sin 55 5 sin 55 3 9.2 sin 70 x 5 sin 55 3 9.2 sin 70 To solve for x, I multiplied oth sides of the eqution y sin 55. I used lultor to evlute. x 8 8.0 m The unknown side of the tringle is out 8.0 m long. 284 5.3 Investigting nd pplying the Sine Lw in ute Tringles
EXMPLE 2 Seleting the sine lw s strtegy to lulte unknown ngles Determine ll the interior ngles in ^PQR. Round your nswers to the nerest degree. Q 2 m 98 m 24 P R Tom s Solution /QPR 5 80 2 24 / QPR nd 24 dd up to 80. 5 56 So, to determine / QPR, I sutrted 24 from 80. sin R 2 5 sin 56 98 ^PQR doesn t hve 90 ngle. So I hose the sine lw, sine I knew two sides nd n ngle opposite one of those sides. 2 3 sin R sin 56 5 2 3 2 98 sin R 5 2 3 sin 56 98 I wrote the rtios with the ngles in the numertors to mke it esier to solve for / R. To solve for / R, I multipled oth sides of the eqution y 2. I used the inverse sine funtion on lultor to evlute. /R 8 7 /Q 5 80 2 /QPR 2 /R 5 80 2 56 2 7 5 53 The interior ngles in ^PQR re 56, 53, nd 7. ll three interior ngles dd up to 80. So I sutrted /R nd /QPR from 80 to determine /Q. hpter 5 Trigonometry nd ute Tringles 285
EXMPLE 3 Solving prolem y using the sine lw wll tht is.4 m long hs strted to len nd now mkes n ngle of 80 with the ground. 2.0 m ord is jmmed etween the top of the wll nd the ground to prop the wll up. ssume tht the ground is level. ) Wht ngle, to the nerest degree, does the ord mke with the ground? ) Wht ngle, to the nerest degree, does the ord mke with the wll? ) How fr, to the nerest tenth of metre, is the ord from the se of the wll?.4 m 2.0 m 80 Iselle s Solution ) ) ).4 3 sin u 5.4 3.4 2.0 The ord mkes n ngle of out 44 with the ground. 80 2 80 2 44 5 56 The ord mkes n ngle of out 56 with the wll..4 m sin u.4 sin u 5.4 3 56 80 5 2.0 u5sin 2.4 3 u 8 44 d 2.0 2.0 m 44 2.0 This tringle doesn t hve 90 ngle. So I hose the sine lw, sine I knew two sides nd n ngle opposite one of those sides. To solve for u, I multiplied oth sides of the eqution y.4. I used the inverse sine rtio on lultor to determine the ngle. The interior ngles dd up to 80. So I sutrted the two known ngles from 80. I lelled the distne long the ground etween the wll nd the ord s d. 286 5.3 Investigting nd pplying the Sine Lw in ute Tringles
sin 56 d sin 56 5 2.0 3 d 5 sin 56 3 2.0 sin 56 d 5 sin 56 3 2.0 d 8.7 m The ord is out.7 m from the se of the wll. Then I used the sine lw. To solve for d, I multiplied oth sides of the eqution y sin 56 nd used lultor to evlute. In Summry Key Ide The sine lw sttes tht, in ny ute ^, the rtios of eh side to the sine of its opposite ngle re equl. sin 5 or sin sin 5 5 sin sin 5 sin Need to Know The sine lw n e used only when you know two sides nd the ngle opposite known side or two ngles nd ny side or hpter 5 Trigonometry nd ute Tringles 287
HEK Your Understnding Y. ) Given ^XYZ t the left, lel the sides with lower se letters. ) Stte the sine lw for ^XYZ. X Z 2. Solve eh eqution. Round x to the nerest tenth of unit nd u to the nerest degree. x ) sin 22 5.6 sin 7 3. ) sin u 5 29.2 sin 65 3. Use the sine lw to lulte to the nerest entimetre nd / D to the nerest degree. ) ) D 8 m 70 35 E 48 8 m F Q PRTISING R 45 80 q 7 m S 4. Given ^RQS t the left, determine q to the nerest entimetre. K 5. rhimedes venue nd ernoulli oulevrd meet t n ngle of 45 ner us stop. Riemnn Rod intersets rhimedes venue t n ngle of 60. Tht intersetion is 2 m from the us stop. ) t wht ngle do Riemnn Rod nd ernoulli oulevrd meet? Round your nswer to the nerest degree. ) How fr, to the nerest metre, is the intersetion of Riemnn Rod nd ernoulli oulevrd from the us stop? us stop ernoulli lvd. 45 rhimedes ve. 2 m 60 Riemnn Rd. 288 5.3 Investigting nd pplying the Sine Lw in ute Tringles
6. In ^, two sides nd n ngle re given. Determine the vlue of / to the nerest degree nd the length of to the nerest tenth of entimetre. ) 5 2.4 m, 5 3.2 m, / 5 28 ) 5 9.9 m, 5.2 m, / 5 58 ) 5 8.6 m, 5 9.4 m, / 5 47 d) 5 5.5 m, 5 0.4 m, / 5 30 7. n isoseles tringle hs two 5.5 m sides nd two 32 ngles. T ) lulte the perimeter of the tringle to the nerest tenth of entimetre. ) lulte the re of the tringle to the nerest tenth of squre entimetre. 8. Solve eh tringle. Round eh length to the nerest entimetre nd eh ngle to the nerest degree. ) ) Q 8 m ommunition Tip To solve tringle, determine the mesures of ll unknown sides nd ngles. 70 35 7 m R 45 80 S ) N d) D 5 m L 28 M 8 m E 48 9 m F 9. Solve eh tringle. Round eh length to the nerest tenth of unit nd eh ngle to the nerest degree. ) ^: 5 0.3, 5 4.4, / 5 68 ) ^DEF: /E 5 38, /F 5 48, f 5 5.8 ) ^GHJ: /G 5 6, g 5 5.3, j 5 3. d) ^KMN: k 5 2.5, n 5 9.6, /N 5 42 e) ^PQR: p 5.2, r 5.6, /R 5 52 f) ^XYZ: z 5 6.8, /X 5 42, /Y 5 77 hpter 5 Trigonometry nd ute Tringles 289
2.8 m 2.9 m 86 2.2 m 0. Toy uses hins nd winh to lift engines t his fther s grge. Two hooks in the eiling re 2.8 m prt. Eh hook hs hin hnging from it. The hins re of length.9 m nd 2.2 m. When the ends of the hins re tthed, they form n ngle of 86. In this onfigurtion, wht ute ngle, to the nerest degree, does eh hin mke with the eiling?. etsy instlled ordless phones in the student entre t points,, nd s shown. Explin how you n use the given informtion to determine whih two phones re frthest prt. 65 289 m 5 2. Tom sys tht he doesn t need to use the sine lw euse he n lwys determine solution y using primry trigonometri rtios. Is Tom orret? Wht would you sy to Tom to onvine him to use the sine lw to solve prolem? Extending 3. Two ngles in tringle mesure 54 nd 38. The longest side of the tringle is 24 m longer thn the shortest side. lulte the length, to the nerest entimetre, of ll three sides. 4. Use the sine lw to show why the longest side of tringle must e opposite the lrgest ngle. P 6.5 m 4 5. tringulr grden is enlosed y fene. dog is on 5 m lesh tethered to the fene t point P, 6.5 m from point, s shown t the left. If / 5 4, lulte the totl length, to the nerest tenth of metre, of fene tht the dog n reh. 6. Tr uilt sulpture in the shpe of huge equilterl tringle of side length 4.0 m. Unfortuntely, the ground underneth the sulpture ws not stle, nd one of the verties of the tringle snk 55 m into the ground. ssume tht Tr s sulpture ws uilt on level ground originlly. ) Wht is the length, to the nerest tenth of metre, of the two exposed sides of Tr s tringle now? ) Wht perent of the tringle s surfe re remins ove ground? Round your nswer to the nerest perent. 290 5.3 Investigting nd pplying the Sine Lw in ute Tringles