UNIT 7 RIGHT ANGLE TRIANGLES

Transcription

1 UNIT 7 RIGHT ANGLE TRIANGLES Assignment Title Wrk t cmplete Cmplete Cmplete the vcabulary wrds n Vcabulary the attached handut with infrmatin frm the bklet r text. 1 Triangles Labelling Triangles 2 Pythagrean Therem Pythagrean Therem 3 Trignmetry Trignmetry 4 The Sine Rati The Sine Rati Angle f Elevatin and Depressin 5 Using Sine Rati in Slving Right Triangles Using Sine Rati in Slving Right Triangles 6 The Csine Rati The Csine Rati 7 Using Csine Rati in Slving Right Triangles Using Csine Rati in Slving Right Triangles 8 The Tangent Rati The Tangent Rati 9 Using Tangent Rati in Slving Right Triangles Using Tangent Rati in Slving Right Triangles 10 Finding Angles Finding Angles 11 Slving Sight Triangles Slving Sight Triangles Mental Math Mental Math Nn-calculatr practice Get this page frm yur teacher Practice Test Practice Test Hw are yu ding? Get this page frm yur teacher Self- Assessment Chapter Test Self-Assessment Traffic Lights Chapter Test Shw me yur stuff! On the next page, cmplete the selfassessment assignment. 1

2 Traffic Lights In the fllwing chart, decide hw cnfident yu feel abut each statement by sticking a red, yellw, r green dt in the bx. Then discuss this with yur teacher BEFORE yu write the test. Statement After cmpleting this chapter; I can use the Pythagrean therem t calculate the missing side f a right triangle I knw when t chse sine (sin), csine (cs) r tangent(tan) based n the infrmatin given I can use the three basic trignmetric functins (sin, cs, tan) t find a missing side r angle f a right triangle Dt I can determine places in the wrkplace where I culd us trignmetry 2

3 Vcabulary: Unit 7 Right Angle Triangles budget *this term has been cmpleted fr yu as an example angle f depressin Definitin an estimate f the amunt f mney t be spent n a specific prject r ver a given time frame Definitin Diagram: A sample f a persnal mnthly budget: Net Pay $2500 Rent $600 Recreatin $100 Telephne $75 Persnal Care $100 Utilities $75 Savings $150 Fd $500 Spending (CDs ) $200 Transprtatin $500 Other expenses $100 Clthing $100 Ttal $2,500 Diagram/Example angle f elevatin Definitin Diagram/Example csine Definitin Diagram/Example hyptenuse Definitin Diagram/Example 3

4 leg Definitin Diagram/Example Pythagrean therem Definitin Diagram/Example right triangle Definitin Diagram/Example sine Definitin Diagram/Example tangent Definitin Diagram/Example 4

5 TRIANGLES In this unit, yu will be lking at triangles, specifically right angle triangles, als called right triangles. Yu will learn abut Pythagrean Therem and the basic trignmetric ratis. But first it is necessary t start with sme facts abut triangles. Fact 1: Every triangle cntains 3 sides and 3 angles r vertices (plural f vertex). Fact 2: The measurements f these angles always ttal Remember this frm the last unit?? Fact 3: T identify the side r vertex in a triangle, it is imprtant t label the triangle fllwing a standard rutine. Each vertex f a triangle is labeled with a capital case letter like A - and each side is labeled with the lwer case letter that matches the ppsite vertex. An example is belw. A c b B a C Anther way t label the sides is with the capital letters f the tw vertices the side cnnects. An example is belw. A c b Side a can be called BC. Side b can be called AC. Side c can be called AB. B a C 5

6 ASSIGNMENT 1 LABELLING TRIANGLES 1) Label each side f the triangles belw using a single lwer case letter matching the ppsite vertex. a) X b) R Y Z S T c) d) D A E F B C 2) Label each vertex f the triangles belw using a single capital letter matching the ppsite side. a) b) f a b d e c c) d) q p r w x y 6

7 PYTHAGOREAN THEOREM Pythagrean Therem states the relatinship between the sides f a right triangle. S, mre facts abut triangles are necessary. Fact 4: A triangle that cntains a 90 0 angle (a right angle) is called a right triangle (r right-angle triangle). Fact 5: The side f the triangle that is ppsite the 90 0 angle is always called the hyptenuse. It is labelled in the triangle belw. The ther tw sides f the triangle are called legs. hyptenuse Fact 6: The hyptenuse is always the lngest side in the triangle. It is always ppsite the largest angle which is the 90 0 r right angle. Fact 7: Pythagrean Therem states that in any right triangle, the sum f the squares f the lengths f the legs is equal t the square f the length f the hyptenuse. S in ABC with the right angle at C, the fllwing relatinship is true: c 2 = a 2 + b 2 where a and b are the ther 2 legs f the triangle. c b a 7

8 Often Pythagrean Therem is illustrated as the square f the sides as fllws: Ntice that the length f side a is 3 bxes, side b is 4 bxes, and side c is 5 bxes. S if we calculate the area f each square, the fllwing is true: c c = c 2 = 5 5 = 25 b b = b 2 = 4 4 = 16 a a = a 2 = 3 3 = 9 And we knw that c 2 = a 2 + b 2 S 25 = which is a true statement! We can als rearrange the equatin t find the length ne f the legs; c 2 = a 2 + b 2 a 2 = c 2 b 2 b 2 = c 2 a 2 When we use Pythagrean Therem t find a length f the hyptenuse r a leg, yu need t have a calculatr that has the square rt functin n it. The cmputer symbl lks like this: r Example 1: Use Pythagrean Therem t find the length f the missing side t ne decimal place. P Slutin: q 2 = p 2 + r cm q q 2 = q 2 = q 2 = Q 5.2 cm R = q 6.44 cm Side q is apprximately 6.4 cm 8

9 Example 2: Use Pythagrean Therem t find the length f the missing side t ne decimal place. A Slutin: c 2 = a 2 + b 2 b 12.8 in S, b 2 = c 2 a 2 b 2 = b 2 = C in B b 2 = = b 6.90 cm Side b is apprximately 6.9 cm ASSIGNMENT 2 PYTHAGOREAN THEOREM 1) Using the fllwing triangles, use lettering prvided t state the Pythagrean relatins that apply. a) b) A X B z y Y x Z C D 2) Rearrange the fllwing Pythagrean statement t slve fr the ther tw legs. d 2 = e 2 + f 2 9

10 3) A ladder is leaned against a huse. The base f the ladder is d feet away frm the huse. Draw a diagram and then write the Pythagrean relatinship that exists fr these lengths. Use l fr the ladder, h fr the huse, and d fr the distance the ladder is frm the huse. Yu are nt required t slve this questin. 4) A 40 ft ladder reaches 38 feet up the side f a huse. Hw far frm the side f the huse is the base f the ladder? Draw a diagram and shw yur wrk. 5) A ramp int a huse rises up 3.5 meters ver a hrizntal distance f 10.5 meters. Hw lng is the ramp? Draw a diagram and shw yur wrk. 10

11 TRIGONOMETRY Trignmetry is ne f the mst imprtant tpics in mathematics. Trignmetry is used in many fields including engineering, architecture, surveying, aviatin, navigatin, carpentry, frestry, and cmputer graphics. Als, until satellites, the mst accurate maps were cnstructed using trignmetry. The wrd trignmetry means triangle measurements. It is necessary t finish ur triangle facts here. Fact 8: In trignmetry, the ther tw sides (r legs) f the triangle are referred t as the ppsite and adjacent sides, depending n their relatinship t the angle f interest in the triangle. In this example, if we pick angle DEF the angle labelled with the Greek letter θ then we are able t distinguish the sides as illustrated in the diagram belw. D ppsite hyptenuse θ F adjacent E The side that is ppsite the angle f interest, in this case θ, is called the ppsite side. The side that is nearest t angle θ and makes up part f the angle is called the adjacent side. T help yu, remember that adjacent means beside. Althugh the hyptenuse ccupies ne f the tw adjacent psitins, it is never called the adjacent side. It simply remains the hyptenuse. This is why it is identified first. It is recmmended t label the side in the rder hyptenuse, ppsite, and finally adjacent. Yu may use initials fr these side, h,, and a, but always use lwer case letters t avid mixing up the labelling with a vertex. 11

12 Example 1: Using the triangle belw, answer the questins θ 12 1) What is the hyptenuse? 2) What is the ppsite side t θ? 3) What is the adjacent side t θ? Slutin: 1) What is the hyptenuse? 15 2) What is the ppsite side t θ? 9 3) What is the adjacent side t θ? 12 This example uses the same triangle as in Example 1; hwever, this time, the ther acute angle is labelled as θ. This is dne t shw that the ppsite and adjacent sides switch when the ther angle is the angle f interest. The hyptenuse always stays the same. Example 2: Using the triangle belw, answer the questins. 15 θ 9 12 Slutin: 1) What is the hyptenuse? 2) What is the ppsite side t θ? 3) What is the adjacent side t θ? 1) What is the hyptenuse? 15 2) What is the ppsite side t θ? 12 3) What is the adjacent side t θ? 9 12

13 ASSIGNMENT 3 TRIGONOMETRY Fr each f the right triangles belw, mark the hyptenuse, and the sides that are ppsite and adjacent sides t θ as shwn in the example. Example: θ h = hyptenuse h = ppsite a = adjacent a 1) 2) θ θ 3) 4) θ θ 13

14 TRIGONOMETRIC RATIOS In the previus unit abut similar figures, yu learned that the ratis f crrespnding sides f similar triangles are equal. When the angles f different triangles are the same, the rati f the sides within the triangle will always be the same. They depend nly n the measure f the angle f interest, nt the size f the triangle. These ratis are the trignmetric ratis. There are three trignmetric ratis we are cncerned with: sine, csine, and tangent. THE SINE RATIO The sine f angle θ means the rati f the length f ppsite side t the length f the hyptenuse. It is abbreviated as sin θ but read as sine θ. It is written like this: ppsite sin θ = hyptenuse r sin θ = h Example 1: Find the sine f θ in this triangle. Rund t 4 decimal places θ 12 Slutin: The ppsite side is 5 and the hyptenuse is 13. S sin θ = h = 13 5 = S sin θ = Nte: Runding t 4 decimal places is standard when calculating trignmetric ratis. Example 2: Use yur calculatr t determine the fllwing sine ratis. Rund t 4 decimal places. a) sin 15 0 b) sin 67 0 c) sin 42 0 ***** REMEMBER TO SET YOUR CALCULATOR ON DEGREES (DEG) **** Slutin: Type sin fllwed by the angle, and then = t slve a) sin 15 0 = b) sin 67 0 = c) sin 42 0 =

15 ASSIGNMENT 4 THE SINE RATIO 1) Calculate the value f sin X t tw decimal places. a) b) X 5.2 in 8.1 in 6.9 m 9.6 in X 4.3 m 2) Use yur calculatr t determine the value f each f the fllwing sine ratis t fur decimal places. a) sin 10 0 = b) sin 48 0 = c) sin 77 0 = d) sin 85 0 = 3) There are tw special sine ratis. Calculate the fllwing and suggest why the values are what the results give yu. a) sin 0 0 = b) sin 90 0 = 15

16 ANGLE OF ELEVATION AND DEPRESSION When yu lk up at an airplane flying verhead fr example, the angle between the hrizntal and yur line f sight is called the angle f elevatin. When yu lk dwn frm a cliff t a bat passing by, the angle between the hrizntal and yur line f sight is called the angle f depressin. When yu are given the angle f depressin, it is imprtant t carefully use this angle in yur triangle. Example 1: Yu are standing at the tp f a cliff. Yu spt a bat 200 m away at an angle f depressin f 55 0 t the hrizn. Hw far is the bat frm the cast? Draw a diagram t illustrate this situatin. Slutin: Draw a diagram, label it with the infrmatin, and then slve the triangle. θ hrizn 55 0 Angle f depressin 200 m The angle inside the triangle is the cmplement t the angle f depressin. T find that angle, d the fllwing: θ = θ = 35 0 x 16

17 USING SINE RATIO IN SOLVING RIGHT TRIANGLES Whenever ne side and ne angle f a right triangle are already knwn, the remaining sides can be fund using the trignmetric ratis. The sine rati can be used t find missing parts f a right triangle. Example 1: Use the sine rati t find the x in the triangle belw. x 9 θ = 35 0 Slutin: Step 1: Label the sides f the triangle with h, and a h θ = 35 0 x a 9 m Step 2: Circle the number with the side it represents and the unknwn (x) with the side it represents. Step 3: Identify the rati required t slve fr x Since and h are being used, the crrect rati is sin θ Step 4: Substitute the crrect values int the crrect rati. sin θ = h sin 35 0 = x 9 Step 4: Slve using the prcess Crss Multiply and Divide. Since sin 35 0 = sin 35 1, then sin sin 35 = becmes x 1 = x 9 x = 9 1 sin 35 0 = = 15.7 m 17

18 Example 2: A ladder 8.5 m lng makes an angle f 72 0 with the grund. Hw far up the side f a building will the ladder reach? Slutin: Sketch a diagram and place the infrmatin frm the questin n this diagram. Remember that there will always be a right triangle in yur diagram. It is ften helpful t draw that triangle and cpy the key infrmatin frm the sketch. h 8.5 m x 72 0 a Step 1: Label the sides f the triangle with h, and a See abve right. Step 2: Circle the number with the side it represents and the unknwn (x) with the side it represents. Step 3: Identify the rati required t slve fr x Since and h are being used, the crrect rati is sin θ Step 4: Substitute the crrect values int the crrect rati. sin θ = h sin 72 0 = x 8.5 Step 4: Slve using the prcess Crss Multiply and Divide. Since sin 72 0 sin 72 =, then sin 72 0 x sin = becmes x = sin = = 8.1 m = x

19 ASSIGNMENT 5 USING SINE RATIO IN SOLVING RIGHT TRIANGLES 1) Calculate the length f the side indicated in the fllwing diagrams. a) b) cm x x 5.2 m ) A weather balln with a 15 m string is tied t the grund. Hw high is the balln if the angle between the string and the grund is 38 0? 3) A ramp makes an angle f 22 0 with the grund. If the end f the ramp is 1.5 m abve the grund, hw lng is the ramp? 19

20 THE COSINE RATIO The csine f angle θ means the rati f the adjacent side t the hyptenuse. It is abbreviated as cs θ but read as csine θ. It is written like this: adjacent cs θ = hyptenuse r cs θ = h a Example 1: Find the csine f θ in this triangle θ 12 Slutin: The adjacent side is 12 and the hyptenuse is 13. S a 12 cs θ = = = h 13 Nte: Runding t 4 decimal places is standard when calculating trignmetric ratis. Example 2: Use yur calculatr t determine the fllwing csine ratis. Rund t 4 decimal places. a) cs 15 0 b) cs 67 0 c) cs 42 0 ***** REMEMBER TO SET YOUR CALCULATOR ON DEGREES (DEG) **** Slutin: Type cs fllwed by the angle, and then = t slve a) cs 15 0 = b) cs 67 0 = c) cs 42 0 =

21 ASSIGNMENT 6 THE COSINE RATIO 1) Calculate the value f cs X t tw decimal places. a) b) X 5.2 in 8.1 in 12.4 cm 7.9 cm 9.6 in X 2) Use yur calculatr t determine the value f each f the fllwing sine ratis t fur decimal places. a) cs 10 0 = b) cs 48 0 = c) cs 77 0 = d) cs 85 0 = 3) There are tw special csine ratis. Calculate the fllwing and suggest why the values are what the results give yu. a) cs 0 0 = b) cs 90 0 = 21

22 USING COSINE IN SOLVING RIGHT TRIANGLES Whenever ne side and ne angle f a right triangle are already knwn, the remaining sides can be fund using the trignmetric ratis. The csine rati can be used t find missing parts f a right triangle. Example 1: Use the crrect trig rati t find the x in the triangle belw. 5 cm θ = 30 0 x Slutin: Step 1: Label the sides f the triangle with h, and a h 5 cm θ = 30 0 x a Step 2: Circle the number with the side it represents and the unknwn (x) with the side it represents. Step 3: Identify the rati required t slve fr x Since a and h are being used, the crrect rati is cs θ Step 4: Write dwn the chsen rati and substitute the crrect values int the crrect rati. a cs θ = h cs 30 0 x = 5 Step 5: Slve using the prcess Crss Multiply and Divide. Since cs 30 0 = cs 30 1, then cs 30 0 x cs 30 = becmes 5 1 = 5 x x = cs = = 4.3 cm 22

23 ASSIGNMENT 7 USING COSINE RATIO IN SOLVING RIGHT TRIANGLES 1) Calculate the length f the side indicated in the fllwing diagrams. a) x b) cm x m 2) A child s slide rises t a platfrm at the tp is If the hrizntal distance that the slide cvers is 25 m lng, hw lng is the slide? ) A flagple is anchred t the grund by a guy wire that is 12 m lng. The guy wire makes and angle f 63 0 with the grund. Hw far frm the base f the flagple must the guy wire be anchred int the grund? 23

24 THE TANGENT RATIO The tangent f angle θ means the rati f the ppsite side t the adjacent side. It is abbreviated as tan θ but read as tangent θ. It is written like this: ppsite tan θ = adjacent r tan θ = a Example 1: Find the tangent f θ in this triangle θ 12 Slutin: The ppsite side is 5 and the adjacent side is 12. S tan θ = a = 12 5 = Nte: Runding t 4 decimal places is standard when calculating trignmetric ratis. Example 2: Use yur calculatr t determine the fllwing tangent ratis. Rund t 4 decimal places. a) tan 15 0 b) tan 67 0 c) tan 42 0 ***** REMEMBER TO SET YOUR CALCULATOR ON DEGREES (DEG) **** Slutin: Type tan fllwed by the angle, and then = t slve a) tan 15 0 = b) tan 67 0 = c) tan 42 0 =

25 ASSIGNMENT 8 THE TANGENT RATIO 1) Calculate the value f tan X t tw decimal places. a) b) X 6.5 m 5.2 in 8.1 in 9.6 in X 5.1 m 2) Use yur calculatr t determine the value f each f the fllwing sine ratis t fur decimal places. a) tan 10 0 = b) tan 48 0 = c) tan 77 0 = d) tan 85 0 = 3) There are sme special tangent ratis. Calculate the fllwing and suggest why the values are what the results give yu. a) tan 0 0 = b) tan 45 0 = c) tan 89 0 = d) tan 90 0 = 25

26 USING TANGENT IN SOLVING RIGHT TRIANGLES Whenever ne side and ne angle f a right triangle are already knwn, the remaining sides can be fund using the trignmetric ratis. The tangent rati can be used t find missing parts f a right triangle. Example 1: Use the crrect trig rati t find the x in the triangle belw. 2 mm θ = 15 0 x Slutin: Step 1: Label the sides f the triangle with h, and a h 2 mm θ = 15 0 x a Step 2: Circle the number with the side it represents and the unknwn (x) with the side it represents. Step 3: Identify the rati required t slve fr x Since and a are being used, the crrect rati is tan θ Step 3: Substitute the crrect values int the crrect rati. tan θ = a tan 15 = x 2 Step 4: Slve using the prcess Crss Multiply and Divide. Since tan 15 0 = tan15 1, then tan tan15 = becmes x 1 = x 2 x = 2 1 tan 15 0 = = 7.5 mm 26

27 ASSIGNMENT 9 USING TANGENT RATIO IN SOLVING RIGHT TRIANGLES 1) Calculate the length f the side indicated in the fllwing diagrams. a) 6.5 cm b) 48 0 x 9.2 m 37 0 x 2) A man stands 12 m frm the base f a tree. He views the tp f the tree at an angle f elevatin f Hw tall is the tree? 3) Hw far frm the side f a huse is the base f a ladder if the angle f elevatin is 70 0 and the ladder reaches 15 feet up the side f the huse? 27

28 FINDING ANGLES S far in this unit, yu have used the trignmetric ratis t find the length f a side. But if yu knw the trignmetric rati, yu can calculate the size f the angle. This requires and inverse peratin. Yu can use yur calculatr t find the ppsite f the usual rati prvided yu can calculate the rati. T d this yu need 2 sides in the triangle. Yu can think f the inverse in terms f smething simpler: additin is the ppsite r inverse f subtractin. In the same way, trig functins have an inverse. T calculate the inverse, yu usually use a 2nd functin and the sin/cs/tan buttns n yur calculatr in sequence. If yu lk at yur calculatr just abve the sin/cs.tan buttns, yu shuld see the fllwing: sin -1, cs -1, tan -1. These are the inverse functins. If yu use these buttns, yu will be able t turn a rati int an angle. Example 1: Calculate each angle t the nearest whle degree. a) sin X = b) cs Y = c) tan Z = Slutin: Use the apprpriate inverse functin n yur calculatr. NOTE: Every calculatr is different in hw the buttns are keyed in rder t achieve the desired utcme. Mst calculatrs will need t key 2ndF sin in rder t get sin -1 displayed. Then key in the value with r withut brackets as necessary. a) sin X = X = sin -1 (0.2546) X = Angle X is b) cs Y = Y = cs -1 (0.1598) Y = Angle Y is c) tan Z = Z = tan -1 (3.2785) Z = Angle Z is

29 Example 2: Determine the angle θ in the fllwing triangle. 5 m θ 3 m Slutin: 1) h,, a the triangle 2) Circle the letters with their partner numbers 3) Chse the apprpriate trig rati. In this case, h 5 m it is tangent. 4) Write dwn the rati and fill it in. tan θ = a θ tan θ = m a 5) Divide the numeratr by the denminatr in the fractin t get a decimal number. tan θ = ) Use the inverse functin t slve fr θ. θ = tan -1 ( ) θ = Angle θ is apprximately

30 ASSIGNMENT 10 FINDING ANGLES 1) Calculate the fllwing angles t the nearest whle degree. a) sin D = b) cs F = c) tan G = d) sin P = e) cs Q = f) tan R = ) In a right triangle, XYZ, the rati f the ppsite side t X t the hyptenuse is 7:8 r 8 7. What is the apprximate size f X? 3) At what angle t the grund is an 8 m lng cnveyr belt if it is fastened 5 m frm the base f the lading ramp? 30

31 4) If a bat is 150 m frm the base f a 90 m cliff, what is the angle f elevatin frm the bat t the tp f the cliff? 5) After an hur f flying, a jet has travelled 300 miles, but gne ff curse 48 miles west f its planned flight path. What angle, θ, is the jet ff curse? 48 mi 300 mi θ 6) What is the angle f depressin, θ, frm the tp f a 65 m cliff t an bject 48 m frm its base? θ 31

32 SOLVING RIGHT TRIANGLES When asked t slve a right triangle, that means t find all the angle measures and the length f all the sides. Remembering that the angles in a triangle add up t 180 0, nce tw f the angles are knwn, the third can be calculated by subtractin. Als, nce tw f the sides are knwn, the third side can be fund using Pythagrean Therem unless tld nt t use it! Then the third side shuld be fund using trig ratis. Example 1: Slve the right triangle belw. Give lengths t the nearest tenth f a cm, and angles t the nearest whle degree. R q = 8.7 cm p 56 0 P r Q Slutin: Subtract t find the third angle, use trig t find side p, and use Pythagrean Therem t find side r. Part 1: R = R= 34 0 Part 2: T slve fr side p, use the sin rati. Use P and the hyptenuse, 8.7 cm sin P = h sin 56 1 = p 8.7 p = sin p = 7.2 cm Part 3: Use Pythagrean Therem t find side r q 2 = p 2 + r = r 2 r 2 = r 2 = = r 4.88 cm r 4.9 cm 32

33 Example 2: Slve the right triangle belw withut using Pythagrean Therem. Give lengths t the nearest tenth f a cm, and angles t the nearest whle degree. X z y = 5.4 m 48 0 Y x Z Slutin: Subtract t find the third angle, use trig t find side x, and side z. Part 1: X = R= 42 0 Part 2: T slve fr side x, use the tan rati. Use Y and the side y, 5.4 m tan Y = a tan 48 1 = 5.4 x x = tan 48 0 x = 4.9 m Part 3: T slve fr side z, use the sin rati. Use Y and the side y, 5.4 m sin Y = h sin 48 1 = 5.4 z z = sin 48 0 z = 7.3 m Nte: any trig rati invlving side z wuld wrk. These numbers were chsen because they are exact frm the given infrmatin, and thus mre accurate. 33

34 Example 3: Slve the right triangle belw withut using Pythagrean Therem. Give lengths t the nearest tenth f a cm, and angles t the nearest whle degree. X z y = 16.3 mi m Y 15.4 mi Z Slutin: Use trig t find Y, subtract t find the third angle, and use trig t find side z. Part 1: T find Y, use the tan rati. Use side x, 15.4 mi and the side y, 16.3 mi tan Y = a 16.3 tan Y = 15.4 tan Y = θ = tan -1 (1.0584) θ = Angle θ is apprximately Part 2: : X = R= 43 0 Part 3: T slve fr side z, use the sin rati. Use Y and the side y, 5.4 m sin Y = h sin 43 1 = 16.3 z z = sin 43 0 z = 23.9 mi 34

35 ASSIGNMENT 11 SOLVING RIGHT TRIANGLES 1) Slve the given triangle. a) D E m F b) Q 135 cm R 200 cm P 35

36 2) Slve the triangle belw withut using Pythagrean Therem. A 6.8 in B 37 0 C 36

37 UNIT REVIEW The techniques used t slve prblems invlving right angle triangles have widespread applicatins in many real-life areas. Often, wrd prblems are presented where Pythagrean Therem r trignmetry ratis are used t answer a questin. The fllwing sectin f questins requires that yu slve these prblems in that way. Remember these pints t help yu slve trig prblems: 1. After reading the given infrmatin carefully, draw and label a diagram f a triangle if ne is nt prvided. 2. When a side is the unknwn and n angle is given, use the Pythagrean therem t find the unknwn side. 3. When a side is the unknwn and an angle is given, use the apprpriate trig rati and nly the sin, cs, r tan key n the calculatr. 4. When an angle is the unknwn, use the 2ndF key in additin t the sin, cs r tan key t get the inverse functins: sin -1, cs -1, tan Label the triangle with, a, and h in rder t chse the crrect trig rati. 6. Slve fr the unknwn. 7. Check t see if yur answer is reasnable. Sme students find it helpful t remember the trignmetric relatinships f sine, csine, and tangent with the phrase SOH CAH TOA (prnunced sw-a caw te-a).this cmes frm the initials frm the trig ratis and their sides: sin = h cs = h a tan = a Yu will be given the trignmetric ratis and Pythagrean Therem in the Data Pages fr bth yur tests and the Prvincial exam. Yu are respnsible fr knwing hw t use and apply these frmulas. 37