ONDENSED LESSON 1.1 Right Tringle Trigonometr In this lesson ou will lern out the trigonometri rtios ssoited with right tringle use trigonometri rtios to find unknown side lengths in right tringle use trigonometri inverses to find unknown ngle mesures in right tringle Suppose ou fl kite. There is strong wind, so the string is pulled tut. You hve mrked the string, so ou know how muh string hs een let out, nd ou n mesure the ngle the string mkes with the horizontl. You n use trigonometri rtio to find the kite s height. In this lesson ou will lern how. Trigonometr reltes the ngle mesures in right tringles to the side lengths. First, rell tht tringles with the sme ngle mesures re similr, nd so the rtios of orresponding sides re equl. In right tringles, there re speil nmes for the rtios. For n ute ngle A in right tringle, the sine of A is Hpotenuse the rtio of the length of the leg opposite A to the length of the hpotenuse. sin A opposite leg A hpotenuse The osine of A is the rtio of the length of the leg djent to A to the length of the hpotenuse. os A djent leg hpotenuse This leg is djent to A. The tngent of A is the rtio of the length of the opposite leg to the length of the djent leg. opposite leg tn A djent leg Red Emple A in our ook, nd then red the emple elow. This leg is opposite A. EXAMPLE Find the unknown length,. 1 5 A Solution You know the length of the side opposite to the 5 ngle, nd ou wnt to find the length of the hpotenuse. Therefore, ou n use the sine rtio. sin 5 1 1 33.13 sin 5 (ontinued) Disovering Advned Alger ondensed Lessons HAPTER 1 175 010 Ke urriulum Press
Lesson 1.1 Right Tringle Trigonometr (ontinued) The inverse of trigonometri funtion gives the mesure of the ngle tht hs given rtio. For emple, sin 30 1_, so sin 1 1_ 30. Emple in our ook uses the inverse tngent funtion. Red the emple refull. Investigtion: Steep Steps Red the opening prgrph of the investigtion in our ook. omplete Steps 1 of the investigtion nd then ompre our nswers to those elow. Step 1 First sketh step with the mimum rise nd minimum run. Let the ngle of inlintion e. euse the floor nd the run re oth horizontl (nd thus prllel), the ngle etween the run nd the hpotenuse is lso. You know the lengths of the opposite nd djent sides, so use tngent to solve for. tn 7.75 10 tn 1 7.75 10 37.8 7.75 in. The ngle of inlintion is out 38. Step Two sets of stirs tht will fit oth the ode nd the rule of thum re set with unit run of 11 in. nd unit rise of 6.5 in. nd set with unit run of 11.5 in. nd unit rise of 6 in. The respetive ngles of inlintion for these sets re given tn 1 6.5 11 30.6 nd tn 1 6 11.5 7.6. An emple of set of stirs tht fits the rule of thum ut does not fit the ode is one with unit rise of 8.75 in. nd unit run of 8.75 in. The ngle of inlintion for this set is given tn 1 8.75 8.75 5.0. Step 3 Refer to the photo nd digrm on pge 68 of our ook.. There re infinitel mn designs possile, ut not ll designs will meet the ode given in Step 1. For emple, stir with unit rise of out 15.6 in. nd unit run of out 1 in. would fit the 0.8 ngle of inlintion ut would not fit the ode, euse the rise is too high.. To find the solution, let the unit run e represented r. Then the unit rise will e represented 17.5 r. To find r, use the tngent rtio. tn 0.8 17.5 r r 0.3799 17.5 r r 0.3799r 17.5 r 1.3799r 17.5 r 17.5 1.3799 r 1.68 in. So the run is 1.68 in. nd the rise is 17.5 1.68.8 in. Step Use the tngent funtion nd let e the ngle of inlintion. Using tn 1 16, tn 1 1 16 3.58 nd using tn 1 0, tn 1 1 0.86. So the ngle should e etween.86 nd 3.58. 10 in. 176 HAPTER 1 Disovering Advned Alger ondensed Lessons 010 Ke urriulum Press
ONDENSED LESSON 1. The Lw of Sines In this lesson ou will disover nd ppl the Lw of Sines, whih desries reltionship etween the sides nd ngles of n olique tringle You hve investigted the reltionships etween the sides nd ngles of right tringles. Now ou will investigte reltionships etween the sides nd ngles of nonright, or olique, tringles. Investigtion: Olique Tringles Step 1 Drw n ute tringle A. Lel the side opposite A s, the side opposite s, nd the side opposite s. Then, drw the ltitude from A to. Lel the height h. At right is one emple. Step From this digrm, ou n write the following equtions: A h sin h, or h sin sin h, or h sin euse oth sin nd sin re equl to h, the re equl to eh other. Tht is, sin sin Dividing oth sides of the eqution ove gives sin sin Step 3 Now, drw the ltitude from to A nd lel the height j. Using method similr to tht in Step, ou should find tht sin A sin (Mke sure ou n derive this eqution on our own!) Steps nd 5 You n omine the proportions from Steps nd 3 to write n etended proportion: sin A sin sin The tringle ou drew in Step 1 ws ute. Do ou think the sme proportion will e true for otuse tringles? Step 6 Drw n otuse tringle A nd mesure eh ngle nd side. A At right is one emple. 3 m Find sin A, sin sin, nd for our tringle. For the tringle t right: sin A sin 31 0.13 sin So, it ppers tht sin A sin sin 3 3 sin 0.13 sin sin 16 0.13 6.3 holds for otuse tringles s well. 31 6.3 m 16 3 m Disovering Advned Alger ondensed Lessons HAPTER 1 177 010 Ke urriulum Press (ontinued)
Lesson 1. The Lw of Sines (ontinued) Emple A in our ook pplies wht ou lerned in the investigtion to rel-world prolem. Red the emple refull. The reltionship ou disovered in the investigtion is lled the Lw of Sines. It is summrized in the Lw of Sines o in our ook. Emple shows how to ppl the Lw of Sines to find n unknown side length in tringle when ou know the mesures of two ngles nd the length of one side. Red the emple refull. Test our understnding finding the length of side A. (Hint: You ll need to find the mesure of first.) You should find tht the length of A is out 15. m. You n lso use the Lw of Sines to find n unknown ngle mesure when ou know two side lengths nd the mesure of the ngle opposite one of the sides. However, in this se ou m find more thn one solution. To help ou understnd wh there m e more thn one solution, look t the digrms on pge 693 of our ook nd red Emple. Here is nother emple. EXAMPLE Solution In A, the mesure of A is 30, the length of side A is 8 m, nd the length of side is 5 m. Sketh nd lel two tringles tht fit this desription. For eh tringle, find the mesures of nd nd the length of side A. The two possiilities re shown elow. A 8 m 5 m 8 m 30 30 A 5 m To find one possile mesure for, use the Lw of Sines. sin 30 5 sin 8 sin 8 sin 30 5 sin 1 8 sin 30 5 53.1 The mesure of is 53.1, so the mesure of is 180 (30 53.1 ), or 96.9. To find the length of A, use the Lw of Sines gin. sin 30 5 sin 96.9 5 sin 96.9 9.9 m sin 30 The length of A is 9.9 m. The other possile mesure for is the supplement of 53.1, or 16.9. The mesure of is then 180 (30 16.9 ), or 3.1. Use the Lw of Sines to find the length of A. sin 30 5 sin 3.1 5 sin 3.1 3.9 m sin 30 The length of A is 3.9 m. 178 HAPTER 1 Disovering Advned Alger ondensed Lessons 010 Ke urriulum Press
ONDENSED LESSON 1.3 The Lw of osines In this lesson ou will use the Lw of osines to find unknown mesures of tringle when ou know two side lengths nd the mesure of the inluded ngle use the Lw of osines to find unknown mesures of tringle when ou know three side lengths You n use the Lw of Sines to find side lengths or ngle mesures of tringle if ou know either two ngle mesures nd one side length or two side lengths nd the mesure of the ngle opposite one of those sides. In Emple A in our ook, ou re given two side lengths nd the mesure of the ngle etween the sides, nd ou must find the length of the third side. The Lw of Sines nnot e pplied in this sitution. Work through the solution to see how to find the unknown side length. If ou use the proedure in Emple A in generl se where ou re given two side lengths, nd, of tringle, A, nd the mesure of the inluded ngle,, ou get the Lw of osines: os where is opposite. Notie tht this looks like the Pthgoren Theorem with n etr term, os. (In ft, if is right ngle, then os is 0 nd the eqution eomes the Pthgoren Theorem.). Red the tet in the Lw of osines o on pge 699 in our ook nd stud the digrms fter the o. Investigtion: Around the orner Red the investigtion in our ook. If ou hve the mterils nd some people to help ou, omplete the investigtion. If not, ou n use the digrm t right. omplete the investigtion on our own, nd then ompre our results to those given. You know the lengths of two sides nd the mesure of n inluded ngle, so ou n use the Lw of osines to find the length of the third side. m 3.5 m A os The Lw of osines..5 (.5)() os 3 Sustitute the known vlues. 6.5 10 os 3 Multipl. 10.5 10 os 3 Solve for. 1.71 Evlute. The two towns re out 1.71 meters prt. (ontinued) Disovering Advned Alger ondensed Lessons HAPTER 1 179 010 Ke urriulum Press
Lesson 1.3 The Lw of osines (ontinued) To find the unknown mesures in Emple, the Lw of osines is pplied twie. Tr to find the unknown mesures ourself, nd then red the solution. In oth the investigtion nd Emple, ou re given two side lengths nd the mesure of the inluded ngle. You n lso use the Lw of osines if ou know three side lengths. The emple elow shows ou how. EXAMPLE Find the ngle mesures. 5.1 m 3.5 m.0 m A Solution Strt using the Lw of osines finding the mesure of. os The Lw of osines. 3.5 5.1.0 (5.1)(.0) os Sustitute the known vlues. 1.5 30.01 0. os Multipl. 17.76 0. os os 17.76 0. Sutrt 30.01 from oth sides. Solve for os. os 1 17.76 0. Tke the inverse osine of oth sides. 9.5 Evlute. Now, use the Lw of Sines to find the mesure of. sin sin 9.5 3.5 sin sin sin.0.0 sin 9.5 3.5 The Lw of Sines. Sustitute the known vlues. Solve for sin. sin 1.0 sin 9.5 3.5 Tke the inverse sine of oth sides. 16.3 Evlute. To find the mesure of A, use the ft tht the sum of the ngle mesures of tringle is 180. A 180 (9.5 16.3 ) 13. Red the reminder of the lesson in our ook, whih summrizes wht ou hve lerned in this nd the previous lesson. 180 HAPTER 1 Disovering Advned Alger ondensed Lessons 010 Ke urriulum Press
ONDENSED LESSON 1. Etending Trigonometr In this lesson ou will etend the definitions of sine, osine, nd tngent to inlude ngles of n mesure find the sine, osine, nd tngent of ngles of rottion use referene ngles to find the sine, osine, nd tngent of relted ngles In Lesson 1.1, the definitions given for sine, osine, nd tngent pplied to ute ngles in right tringles. In this lesson, ou will etend the definitions to ppl to n size ngle. Rememer tht ngles in the oordinte plne re mesured strting from the positive -is nd moving ounterlokwise through Qudrnts I, II, III, nd IV. II III I IV Investigtion: Etending Trigonometri Funtions Red the Proedure Note nd stud the emple shown for Step 1. Then work through the investigtion in our ook. After ou re finished, ompre our nswers to the results elow. Mke sure our lultor is set to degrees. Step 1 The smple nswers use the point (, 0) s the strting point for eh ngle. Your nswers for the oordintes nd the length of the segment will vr depending on the strting point ou hose, ut our results for the sine, osine, nd tngent should mth these results.. 135 sin 135 0.707, os 135 0.707, nd tn 135 1. The oordintes of the rotted point re out (.8,.8). The length of the segment is out (.8).8 3.96 units.. 10 sin 10 0.5, os 10 0.866, nd tn 10 0.577. The oordintes of the rotted point re out ( 3.5, ). The length of the segment is out ( 3.5) ( ).03 units (ontinued) Disovering Advned Alger ondensed Lessons HAPTER 1 181 010 Ke urriulum Press
Lesson 1. Etending Trigonometr (ontinued). 70 sin 70 1, os 70 0, nd tn 70 is undefined. The oordintes of the rotted point re (0, ). The length of the segment is 0 ( ) units. d. 30 sin 30 0.63, os 30 0.766, nd tn 30 0.839. The oordintes of the rotted point re out (3.1,.6). The length of the segment is out 3.1 (.6).05 units. e. 100 sin 100 0.985, os 100 0.17, nd tn 100 5.671. The oordintes of the rotted point re out ( 0.7, 3.9). The length of the segment is out ( 0.7) ( 3.9) 3.96 units. Step The results re summrized elow. From these results ou might hpothesize tht sine is -oordinte -oordinte, osine is -oordinte, nd tngent is segment length segment length Angle Sine osine Tngent 135.8 3.96 10.03 0.71.8 0.50 3.5.03 70 1 0 30.6.05 100 3.9 3.96 0.6 3.1.05 0.98 0.7 3.96 3.96 0.71.8.8 1 0.87 0.57 3.5 0 is undefined 0 0.77.6 0.8 3.1 0.18 0.7 3.9 5.6 -oordinte. (ontinued) 18 HAPTER 1 Disovering Advned Alger ondensed Lessons 010 Ke urriulum Press
Lesson 1. Etending Trigonometr (ontinued) Step 3 ( 3, 1) The length of the segment is ( 3) 1 10. Using the method from Step, 1 sin A, os A 3 10, nd tn A 1 10 3. The lultor gives sin 1 1 10 18.3 nd tn 1 1 3 18.3. This ngle is in Qudrnt I, so it doesn t mth the digrm. However, using the lultor, os 1 3 10 161.57. This ngle ppers to mth the digrm. Step The definitions re loted in the definition o on pge 707 of our ook. Red these definitions refull. Red the prgrph efore Emple A, nd then work through Emples A nd in our ook. If ou need to review speil right tringles, red Refreshing Your Skills for hpter 1 in our ook. elow is nother emple similr to Emple A. EXAMPLE Solution Find the sine, osine, nd tngent of 150 without lultor. Rotte point ounterlokwise 150 from the positive -is. The imge of the point is in Qudrnt II, 30 ove the -is. The referene ngle is 30. The sine, osine, nd tngent of 30 referene ngle re, respetivel, 1_, 3, nd 1 3. 30 150 euse the -oordinte is negtive nd the -oordinte is positive in Qudrnt II, sin 150 1 3, os 150, nd tn 150 1 3. Disovering Advned Alger ondensed Lessons HAPTER 1 183 010 Ke urriulum Press
ONDENSED LESSON 1.5 Introdution to Vetors In this lesson ou will understnd vetors s direted distnes represent ddition, sutrtion, nd slr multiplition of vetors use vetors to solve prolems onvert vetors from one form to nother Some quntities, suh s distne, veloit, nd elertion, n hve diretions ssoited with them. These direted quntities n e represented vetors, whih n e thought of s direted line segments. The line segment hs length, lled the mgnitude, nd diretion. You n represent vetors s segment with n rrowhed t one end, lled the hed or tip. The til is the other end of the vetor. Vetors n e represented in severl ws. The polr form of vetor gives the mgnitude nd the ngle the vetor mkes with the positive -is. For emple, 3 150 represents vetor 3 units long direted 150 ounterlokwise from the positive -is. The retngulr form of vetor gives the horizontl nd 3 3 vertil hnge from the til to the hed. For emple,, 3 represents 3 3 horizontl hnge of nd vertil hnge of _ 3. Equivlent vetors hve the sme mgnitude nd diretion, no mtter where 3 3 the re loted in the oordinte plne. 3 150 nd, 3_ re equivlent vetors. The investigtion eplores some of the properties of vetor ddition nd sutrtion. Note tht _ nd re two ws to designte vetor. In the eqution, the oldfe letters,, nd represent vetors, nd is the resultnt vetor of the lultion. Investigtion: Vetor Addition nd Sutrtion Work through the whole investigtion in our ook, nd then ompre our results to those elow. Steps 1 3 6 0 6 The retngulr form of is 6,. Step i. 6 ii. 5 d 3 e 0 6 3 The retngulr form of is 6,. The retngulr form of is 0, 1. (ontinued) Disovering Advned Alger ondensed Lessons HAPTER 1 185 010 Ke urriulum Press
Lesson 1.5 Introdution to Vetors (ontinued) iii. 3 iv. 6 f 5 3 0 e 6 The retngulr form of is 3, 1. The retngulr form of is 5,. Step 5 If 1, nd 1,, then the sum is 1, 1, 1 1,. Step 6 i. 3 3 3 3 ii. 3 5 3 iii. e 3 d e d 5 iv. 3 e f e f 3 3 Step 7 If 1, nd 1,, then the differene is 1, 1, 1 1,. Step 8 If 1, nd k is slr, then the produt k is k 1, k 1, k. Step 9 The mgnitudes of nd re 3 13 nd 1 17. If 1,, then the mgnitude of, denoted, is 1. Vetors re useful for representing motion. Red Emple A to eplore n pplition of vetor ddition. Sometimes the polr form of vetor is more pproprite. Emple eplins how to onvert from retngulr form to polr form. Red Emple nd mke sure ou understnd how to onvert from retngulr to polr form. Red the tet following Emple. e sure ou understnd how to hnge ering to n ngle tht gives diretion of vetor in polr form. In Emple, the vetors must e onverted from polr form to retngulr form to dd them. Work refull through Emple. 186 HAPTER 1 Disovering Advned Alger ondensed Lessons 010 Ke urriulum Press
ONDENSED LESSON 1.6 Prmetri Equtions In this lesson ou will use prmeter to write prmetri equtions tht seprtel define nd grph prmetri equtions use prmetri equtions to model rel-world prolems So fr, ou hve used equtions to relte nd to eh other. Sometimes ou wnt to epress nd s seprte funtions of third vrile, t, lled the prmeter. These prmetri equtions provide ou with more informtion nd etter ontrol over wht points ou plot. You n use prmetri equtions to epress - nd -oordintes s funtions of time. Emple A in our ook shows how to use prmetri equtions to model motion prolem. Red Emple A nd its solution refull. Then red the following emple. EXAMPLE A Solution Jmes is rowing ot 30 ft ross river. He rows t rte of 1 ft/s diretl towrd the opposite shore. The urrent moves perpendiulr to his diretion of rowing t rte of 3 ft/s. The post where Jmes wnts to tie up his rowot is 100 ft downstrem from his strting point. Will Jmes mke it to the other side of the river efore he psses the post? Let represent the distne in feet the ot moves due to the urrent, let represent the distne in feet Jmes hs rowed ross the river, nd let t represent the time in seonds. Then 3t nd t. Grph this pir of equtions on our lultor. See lultor Note 1 to lern how to enter nd grph prmetri equtions. Use n pproprite window for the ontet. You n piture the post t the point (100, 30). If ou tre point on the grph, ou will see tht Jmes will hve 10 feet to spre efore he rehes the post. Prmetri equtions n help ou model omplited situtions involving motion. Mn pirs of prmetri equtions n e written s single eqution using onl nd. If ou rewrite prmetri model s single eqution, then ou ll hve two different ws to stud sitution. (ontinued) Disovering Advned Alger ondensed Lessons HAPTER 1 187 010 Ke urriulum Press
Lesson 1.6 Prmetri Equtions (ontinued) Investigtion: Prmetri Wlk Steps 1 nd Red Steps 1 nd nd the Proedure Note of the investigtion in our ook. Mke sure ou n visulize wht is going on: A segment is mrked on oordinte grid. As person wlks long the segment, one motion sensor (held reorder X) is reording how the -oordinte of the person s pth hnges nd one sensor (held reorder Y) is reord ing how the -oordinte of the person s pth hnges. Enter the smple dt in our lultor nd omplete the rest of the investigtion on our own. Then ompre our results to those elow. Step 3 Use our lultor to find the medinmedin lines. The medin-medin line for the (t, ) dt is ˆ 0.18t 1.8. Dt olleted reorder X t 0.1 1.78 0.6 1.71 1.1 1.6 1.6 1.50.1 1.3.6 1.36 3.1 1.6 3.6 1.19.1 1.10.6 1.00 5.0 0.95 Dt olleted reorder Y t 0.1 1.95 0.6.0 1.1.11 1.6.1.1.1.6.5 3.1.3 3.6.38.1.39.6.7 5.0.50 Step The medin-medin line for the (t, ) dt is ŷ 0.10t 1.98. Step 5 The grph t right shows plot of the (, ) vlues, long with grphs of the prmetri funtions 0.18t 1.8 nd 0.10t 1.98. The prmetri funtions seem to fit the dt. 1.8 Step 6 Solving ˆ 0.18t 1.8 for t gives t 0.18. Sustitute this epression for t into the eqution for : ŷ 0.1 1.8 0.18 1.98. Step 7 The grph t right shows the (, ) dt nd the funtion ŷ 0.1 1.8 0.18 1.98 from Step 6. Step 8 Eliminting the prmeter gives the sme grph, ut ou lose the informtion out the time, nd ou nnot limit the vlues of t to show onl the segment on the line tht ws tull wlked. (ontinued) 188 HAPTER 1 Disovering Advned Alger ondensed Lessons 010 Ke urriulum Press
Lesson 1.6 Prmetri Equtions (ontinued) Red the tet following the investigtion nd Emple. Emple eplins how to model projetile motion prmetrill. The emple tht follows lso onerns projetile motion. EXAMPLE Peter punts footll t n ngle of 55 so tht it hs n initil veloit of 75 ft/s. If his foot ontts the ll t height 3.5 ft ove the ground, how fr does the ll trvel horizontll efore it hits the ground? Solution Drw piture nd find the - nd -omponents of the initil veloit. os 55 75 sin 55 75 75 os 55 75 sin 55 The horizontl motion is ffeted onl the initil speed nd ngle, so the horizontl distne is modeled 75t os 55. The vertil motion is ffeted the fore of grvit nd the initil height. Its eqution is 16t 75t sin 55 3.5. To find when the ll hits the ground, find t when is 0. 16t 75t sin 55 3.5 0 t 75 sin 55 (75 sin 55 ) ( 16)(3.5) ( 16) t 0.056 or t 3.896 75 ft/s 55 Onl the positive nswer mkes sense in this sitution. The ll hits the ground out 3.896 seonds fter it is kiked. To find how fr the ll hs trveled, sustitute this t-vlue into the eqution for : 75(3.896) os 55 167.6. The ll trvels out 167.6 ft, or 56 d, horizontll. Disovering Advned Alger ondensed Lessons HAPTER 1 189 010 Ke urriulum Press