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Chapter 1 - ClassNet

1. Length of a Include data from five classmates in your table 3 Length of b in the other columns Length of c 4 The ratio z compares the length of a ue i eo l Measure of ZB to the tangent or the e opposite a EEE E _0 C 2 What do the ratios and compare a tanA 5 What do the ratios _ and compare b cos A cos B cos C tan B C 6 What do the ratios a and compare tan C sin A sin B sin C o 7 Record the ratios in questions 4 5 and 6 ad b to 3 decimal places in your table can Bese 8 For which ratios are the values close cae a sin A 9 The ratio A compares the sine of ZA to the value of a What do the ratios nk and 22 compare ae i C sin C 10 Calculate the ratios in question 9 Enter the data to 3 decimal places in your table What do you notice about the values Reflect The sine law shows this relationship for acute triangles sin B How does your investigation verify the sine law Does your investigation show a relationship for cosines or for tangents Justify your answer Did you use The Geometer s Sketchpad or a table What is an advantage or disadvantage of this tool 22 CHAPTER 1 Trigonometry Navigation involves determining the position of a ship or aircraft and plotting its course In navigation direction is often represented using a bearing The bearing is given as a three digit angle measured clockwi
2. 47 55 3633 ZB 77 64 The measures of the other two angles are 55 and 78 to the nearest degree 6 In each expression what is the measure of ZC to the nearest degree 13 sin 83 19 sin 50 i b sin C E 8 4 sin 72 c sin C F a sin C 7 Determine the measures of the angles in each triangle to the nearest degree a B b F __ 264mm c X 370 mm 26 CHAPTER 1 Trigonometry 8 Choose one triangle in question 7 Describe how you calculated the angles 9 Assessment Focus A surveyor has surveyed a triangular plot of land One side of the plot lies along a county road This side is 400 ft long An adjacent side makes an angle of 88 with the side along the road The side opposite this angle is 550 ft long a Determine the length of the third side of the plot to the nearest foot b Describe how you calculated the angles Explain whether your strategy was reasonable 10 Take It Further Amanjeet is flying a kite in a flat open field She lets out 15 m of string The angle of elevation of the kite from her position is 64 Her friend Stephanie is also playing in the field facing Amanjeet The angle of elevation of the kite from Stephanie s position is 35 a Sketch a triangle using the given information b How far apart are the girls to the nearest metre In Your Own Words The measures of two angles in a triangle are known Suppose you want to
3. an angle in a triangle and the ratios of the triangle s side lengths 1 1 Sides of Right Triangles 7 Connect the Ideas The primary trigonometric ratios are sine cosine and tangent You can use these trigonometric ratios to calculate the length of a side of a right triangle if you know the measures of one acute angle and one side a When ZA is an acute angle in a right triangle then An acute angle is less than 30 _ length of side opposite ZA sin A length of hypotenuse P hypotenuse _ length of side adjacent to ZA cos A l length of hypotenuse opposite adjacent PEE oe length of side opposite ZA C length of side adjacent to ZA A 10 ft tall fire truck has an aerial ladder mounted on it The ladder can extend to 101 ft The angle of elevation of the ladder is 63 How far up the building can the ladder reach to the nearest foot From the diagram the hypotenuse is 101 ft Determine the side length b Since we know the length of the hypotenuse c use the sine ratio vertex is labelled with a capital length of side opposite ZB letter Each side is labelled with the rengt OF side Opposite lt D sin B length ar hypotenuse ase letter of the opposite vertex Substitute ZB 63 c 101 sin 63 Multiply each side by 101 sin 63 X 101 4 X 101 sin 63 X 101 b Press 63 101 b 89 99 The base of the ladder is on top of the truck which is 10 ft ET
4. 1 1 1 Name the trigonometric ratio that can be 2 4 2 3 used to calculate each length Explain each choice Then calculate the lengths a X In 1966 the Falls Incline Railway opened in Niagara Falls It moves people from upper parking areas down to Queen Victoria Park The incline track is 170 ft long and has an angle of inclination of 36 What is the vertical distance between the top station and the bottom station Determine the measure of the angle named Explain your method a ZA A C 29cm 14cm Ue b ZX X Y 17 12 Z 4 Simon measures the angle of elevation from the ground to the top of an 18 m high wind turbine He sets a clinometer on the ground 25 m from the base of the tower What is the angle of elevation Pa i 25m 1 3 5 Determine the measure of ZA 1 4 a A 4 7m C 7 2m B A c b 10 2 cm 8 3 cm B a C Mid Chapter Review 29 A welder needs to cut a triangular shape from a piece of metal He knows the lengths of the sides but not the angle measures He needs to know the angles The sine law cannot be used if no angles are known Another relationship relating the sides and angles is needed Inquire Relating Sides and Angles in Triangles Choose Using The Geometer s Sketchpad or Making a Table Using The Geometer s Sketchpad You will need The Geometer s Sketchpad gt Ifyou are using the file Triangle2 gsp
5. b The two longest edges of the kite have the same length Determine this length Take It Further Bogdan and Chloe are standing 70 ft apart on opposite sides of the base of a lighthouse The angle of elevation from each person to the top of the lighthouse is 37 from Bogdan and 56 from Chloe a How tall is the lighthouse b Explain the strategy you used to solve this problem In Your Own Words 44 Tom says he always uses the sine law to determine unknown measures because it is easier to use Amber says you can t use the sine law in every problem Who is correct Explain how you would convince the other person CHAPTER 1 Trigonometry Chapter Review What Do Need to Know Primary Trigonometric Ratios Sine Cosine and Tangent For ZA in a right triangle mAs length of side opposite ZA B length of hypotenuse py ee length of side adjacent to ZA length of hypotenuse ne length of side opposite ZA length of side adjacent to ZA A C The Sine Law In any acute AABC To use the sine law you must know c e the lengths of two sides and the measure of an angle b opposite one of these sides or e the measures of two angles and the length of any side A The Cosine Law In any acute AABC C g b 2ab cos C b e eet Oa E 2ab To use the cosine law you must know e the lengths of two sides and the measure of the angle between them or e the lengths of all three sides C
6. 2 Which imperial unit would you use for each measure Explain your choice a Length of your classroom b Length of a sub sandwich c Penalty spot distance in soccer d Length of the St Lawrence Seaway 3 Roger is replacing the liner of a chimney The length of liner required is 33 ft Roger has 400 in of stainless steel pipe for the liner Is this enough Justify your answer 4 Marlene uses a 2 ft by 1 ft rectangular tray to hold her potted plants The diameters of the top of the pots are 4 and 6 a How many 4 pots can fit on this tray How many 6 pots fit b Marlene puts five 4 pots on one tray How many 6 pots fit in the remaining space Show your work using a diagram Activate Prior Knowledge 3 A ratio compares two quantities Example AABC and ADEF are similar triangles In AABC a 16 cm b 12 cm and c 20 cm In ADEF d 12 cm e 9 cm and f 15 cm Determine the ratio for each pair of corresponding sides Are the ratios equivalent Justify your answer p alled with a capital a a d b b e o cf belled with the of the opposite vertex Solution a ad 16 12 4 3 b bre 12 9 4 3 c cf 20 15 4 3 The ratios are equivalent because each ratio is equivalent to 4 3 J Check 1 ARST and AWXY are similar triangles Determine each ratio a r w b s x c fy R W 2 a What do you notice about the ratios for the corresponding sides in question 1 b
7. Make a prediction about the ratios for the corresponding sides of similar triangles Justify your prediction 3 For the Canadian flag the ratio of width to length is 1 2 What is the perimeter of a Canadian flag that is 12 ft long Are all Canadian flags similar shapes Explain using a diagram 4 CHAPTER 1 Trigonometry To solve an equation for a variable perform the same operation on each side of the equation to isolate the variable Example Solve for x in each equation a 6x 12 b a I c 4 2 3 5x Solution a 6x 12 Divide each side by 6 b 12 Multiply each side by 5 6x 12 X 7 x5 12 X35 ren x 60 c 4 2 3 5x Divide each side by 3 5 4 2 4 2x KA 3 5 1 Solve for x in each equation a 20 8x b 9 c 39 0 75x ajs 2 George spent 18 72 copying a story he wrote Photocopying costs 0 08 per page How many pages are in George s story Use an equation to solve this problem 3 Raja has been offered two jobs Each of these jobs takes 24 weeks to complete One job pays 3440 every 8 weeks The other job pays 2700 every 6 weeks Raja wants to accept the job that pays more per week Show how to use equations to help Raja make her choice 4 Rashad pays 6 30 tax for a pair of shoes that is priced at 45 His friend Zoe purchases another pair of shoes on sale for 25 in the same store How much tax does Zoe have to pay Solve the problem in two ways Show your work Wha
8. create a banner equal to the length of the gym They measure the length of the gym as 2600 cm The fabric is sold by the metre What length of fabric do they need 4 Katie uses a square sheet of paper with side length 7 cm amp amp gt eq E to roll up 50 pennies Each penny is 1 45 mm thick Can Katie roll up all the pennies Justify your answer Loco amp CHAPTER 1 Trigonometry Imperial Units of Length ea The imperial units of length are still in use in many industries Inch in foot ft yard yd and mile mi are units of length in the imperial system 1 ft 12 in 1 yd 3 ft 1 mi 1760 yd imi 5280 ft UE anega TEEPE AEC e EEE A symbol for foot is and for inch is 0 in 1 2 3 A height of 6 ft 6 in is written 6 6 Example From a blueprint a carpenter calculated the length of baseboard needed for a room The perimeter of the room excluding the doorway is 532 in What length of baseboard sold by the foot should the carpenter purchase Solution Since 1 ft 12 in divide 534 by 12 to convert to feet 532 12 44 R4 So 532 in 44 ft 4 in The carpenter should purchase 45 ft of baseboard 1 Convert each measure to the unit indicated he length is rounded an 44 ft 4 in a 6 to inches b 96 yd to feet c 7 mi to feet d 36 to yards e 8 ft 4 in to inches f 3 yd 1 ft to feet g 7040 yd to miles h 192 in to feet i 5 mi to yards
9. determine the length of one side What additional information do you need to use the sine law Illustrate your answer with an example Identify three places after this section where you can find out more about the sine law 1 4 Applying the Sine Law 27 Home Quit Triangle Challenge Materials e triangle cards e scientific calculators Play in a group of 2 or 3 gt Shuffle the cards Place them face down in a pile gt Each player takes a card The player whose triangle shows the longest side length goes first gt Replace the cards Shuffle again and place them face down in a pile Each card shows a gt The first player takes the top card from the pile triangle with some measures labelled but one angle or side to be He or she says what side or angle is to be determined and places it face up determined gt Each player determines that measure gt Players compare answers If the player who turned over the card has the correct answer he or she keeps the triangle card If the player is not correct the triangle card is returned to the bottom of the pile Me hich rule you gt The players take turns displaying cards from the pile ames user ending gt The game ends when each card in the pile the game before you start playing has been used once or when there are no cards left in the pile The player with the most cards is the winner 28 CHAPTER 1 Trigonometry Mid Chapter Review
10. from the north end of the gorge She is 1 9 km from the south end of the gorge The surveyor uses a transit to measure the angle between the two lines of sight as 73 Discuss these questions with a partner gt Which line segment represents the length of the gorge gt Can you use the sine law to determine this length Why or why not gt Can you use the cosine law to determine the length of the gorge Why or why not gt On your own determine the length of the gorge Then compare strategies and solutions with your partner gt Explain your strategy for solving the problem gt Are your results reasonable How do you know 1 6 Applying the Cosine Law 33 Connect the Ideas The cosine law N The cosine law relates the sides and angles in a triangle In any AABC C bB The square of the sum of the twice the the cosine any side equals squares of the minus product of times of the angle other two sides these sides between them The towns of Rockcliff and Sutton are separated by a mountain A straight railway tunnel is to be built joining the towns Rockcliff R is 7 1 km from Treyford T Sutton S is 6 9 km from Treyford The angle between the lines joining each town to Treyford is 40 What is the distance between Rockcliff and Sutton to the nearest tenth of a kilometre In ARST you know the values of rand s and the measure of the angle between them ZT Determine the value
11. ground A 3 5m B In Your Own Words Press I2 B 131 Ei What does the number in the display represent Illustrate your answer with a diagram 18 CHAPTER 1 Trigonometry Many real world situations involve triangles that do not contain right angles The primary trigonometric ratios only apply to right triangles However there are other relationships between the sides and angles of triangles Inquire iad Relating Sides and Angles in Triangles Choose Using The Geometer s Sketchpad or Making a Table Using The Geometer s Sketchpad You will need The Geometer s Sketchpad gt Ifyou are using the file Triangle gsp open it and begin at question 9 The triangle on your screen should look like the one at the right gt If you are not using the file Triangle gsp complete all the questions 1 Start the program A From the Edit menu select Preferences c Set Angle to be measured in degrees to the nearest unit Set Distance to be measured in centimetres to the nearest hundredth b 2 Use the Point tool Click on the screen in three places to construct i three points Use the Text tool A Click on each point to label the points A B and C 3 Use the Selection Arrow tool to select points A and B ki From the Construct menu select Segments Use the Text tool Double click on line segment BC and label it a Double click on line segment AC and label it b Double click on line s
12. is 320 km from its destination D Because of severe weather conditions the plane is diverted to an airport A 250 km from the original destination D The pilot changes his bearing from 060 to 103 How far must the plane fly to reach the airport A Explain how you decided which law to use Sometimes you need to use more than one trigonometric law to solve a problem This is often true if two triangles are involved Example Solution Remember the order of operations Round the value of x to 1 decimal place after calculating Determine the values of x and y C 5 1 cm D y B Use the cosine law to determine the value of x a b 2bc cos A Substitute a x b 5 1 c 4 8 ZA 35 x 5 1 4 8 2 X 5 1 X 4 8 X cos 35 x 8 9443 Take the square root of each side to determine x Vx 8 9443 x 2 99070 To determine the value of y use the sine law c d sin C sin D Substitute c y d x 2 9907 ZC 67 ZD 41 Multiply both sides by sin 67 sin 67 sin 41 Y Xang 2 9907 x sin 67 sin 67 sin 41 y 2 2307 X sin 67 sin 41 y 4 196 The value of x is about 3 0 cm and the value of y is about 4 2 cm 1 7 The Sine Law and the Cosine Law 43 7 Determine the measures of CBD ZCDB side d and side b C 57cm 21cm D 46cm B 8 Jaime s kite looks like this a What is the width of the kite
13. of t Write the cosine law in terms of r s t and ZT t r s 2rs cos T Substitute r 6 9 s 7 1 ZT 40 6 9 7 17 2 X 6 9 X 7 1 X cos 40 t 22 963 Take the square root of each side to determine t JE 422 963 t 4 79 The tunnel will be about 4 8 km long 34 CHAPTER 1 Trigonometry Round angle measures to the nearest degree in your answers 1 Identify the triangle with the unknown length that can be determined using each formula c0 amp b c 2becosA iii a b c Explain how you know the formula you chose is correct a b 2ab cos C b b a c 2ac cos B i B ii A B 7m a 13 ft b eoem C C B C 11 ft A 10m 30cm A 2 Write the formula you can use to determine the length of the unknown side in each triangle Explain how you know your formula is correct n i 4 4m Q 7 1 cm M i 3 4 cm 5 6 0km p a 3 8m C 7 1km K R 3 Determine the length of the unknown side in each triangle b E c P inem 1 20 m 42 cm 1 6 Applying the Cosine Law 35 4 A hiker walked 4 9 km due north from her campsite N Then she walked a further 5 8 km on a bearing of 110 How far from the campsite is she now 110 5 8km 4 9 km When you know the lengths of three sides of a triangle you can use the cosine law to determine the measure of any angle in the triangle Example Determine the measure of ZA in AABC to the nearest degree A oe
14. 2 cm 45 59 F d E A 4 Choose one triangle in question 3 Explain how you calculated the side lengths i en 5 A landscape gardener plans a garden that includes aoe Segia a bridge over a small pond He marks the points on the shore where he wants the bridge to begin and end A and B He marks another point on the shore 5 m from one end of the bridge He measures the angles between the lines of sight joining the points a What is the distance across the pond AB that the bridge must span b The landscape gardener can order prefabricated bridges that span 5 25 m 7 25 m or 9 25 m Which size of bridge should he order Explain 1 4 Applying the Sine Law 25 You have used the sine law to determine the side length of a triangle You can also use the sine law to determine the measure of an angle Example What are the measures of the other A two angles in this triangle to the nearest degree 3 2 cm B 3 6 cm Solution The measure of one angle and the lengths of two sides are given One of the given sides is opposite the given angle Use the sine law 24 smB sinc e a b c Substitute a 3 6 c 3 2 ZC 47 f sinA sinB sin47 pgquate the 1st and 3rd ratios 3 6 b 3 2 A ar Multiply each side by 3 6 e sinA X 36 sin47 X 36 3 6 32 a A sin 47 X 3 6 ond sin A Press 2 4 SIN SIN 47 D x 3 6 3 2 D ZA 55 3633 ZB 180
15. N above the ground So add 10 ft to the height of the ladder 89 99 10 99 99 The ladder can reach about 100 ft up the building 8 CHAPTER 1 Trigonometry Give side lengths to the same number of decimal places as the given lengths 1 3 For each triangle name these sides for the marked angle in one way with two capital letters and in another way with one lower case letter a hypotenuse b opposite c adjacent i B ii C X A a of a triangle can be using the letters that identify its end points For b example in ABC side c can be C Z a For each triangle identify the hypotenuse opposite side and adjacent side for the angle whose measure is given b For each triangle which trigonometric ratio can you use to calculate the length of the indicated side Explain your choice Then use the ratio to calculate the length i 2 7m ii F iii M C oie 49 in 23cm g K g m H G ey N E Draw each triangle then calculate a In ADEF ZD 57 ZF 90 and f 150 ft Determine e b In AXYZ ZY 41 ZZ 90 and z 4 5 m Determine y c In AABC ZB 27 ZC 90 and a 35 cm Determine b Top of tree Ava s town is having a contest to find the tallest tree To measure the height of a pine tree on her family s farm Ava walks 15 0 m from the base of the tree She measures the angle of elevation from the ground to the top of the tree as 65 Ava bas Ground How tal
16. St 8 Determine the measure of the least angle in a triangle whose sides are 6 in 7 in and 9 in 1 7 For each of questions 9 to 11 explain how you decided which law to use 9 A bridge KL is to be built across a river Point M is 50 m from L ZL is 85 and ZM is 35 How long will the bridge be to the nearest metre 10 A gold mine has two ventilation shafts that start at the same work area below ground The vents of the shafts at ground level are 3 2 km apart One shaft is 2 1 km long and the other is 2 5 km long a What is the angle of depression of each shaft b How far below ground is the work area Shaft vent Shaft vent Ground Work area 11 To measure the length of a glacier a geographer stands at a point where she can see both ends of the glacier The angle between the lines of sight from where she stands to the ends of the glacier is 62 She measures the distance from where she stands to the ends to be 250 m and 215 m How long is the glacier Chapter Review 47 Practice Test Multiple Choice Choose the correct answer for questions and 2 Justify each choice 1 Which ratio can you use to determine ZA B 5 cm Sen A 4 cm C A sin A B sin A C cosA D tan A 2 4 5 3 2 Which ratio can you use to determine the length of a B a 20 B a E sin 69 sin 35 sin 35 sin 69 a ssn 20 D a a 28 sin 69 sin 35 sin 35 sin 69 Show al
17. What You ll Learn To determine the measures of sides and angles in right triangles and acute triangles and to solve related problems Trigonometry And Why Real world situations in surveying navigation construction and design can involve determining the measures of sides and angles in triangles 2 CHAPTER 1 Metric Units of Length The metre is a unit of length in the metric system The metric system of units is based on powers of 10 Doaa aiaa aaa Gaa IA aa Oa lcm 10mm 1m 100cm 1km 1000 m lmm 0 lcm lcm 0 01m 1m 0 001 km 0 cm 3 5 6 Example The longest cartoon strip in Guinness World Records 2006 was made by a group of 596 artists The strip measures 238 4 m a What is this length in centimetres b What is this length in kilometres Solution a Since 1 m 100 cm multiply by 100 b Since 1 km 1000 m divide by to convert to centimetres 1000 to convert to kilometres 238 4 X 100 23 840 238 4 1000 0 2384 The cartoon strip is 23 840 cm long The cartoon strip is 0 2384 km long 4 Convert each measure to the unit indicated a 4560 m to kilometres b 156 cm to metres c 2 5 km to metres d 7 2 cm to millimetres e 34 5 m to kilometres f 9 81 mm to metres 2 Which metric unit would you use for each measure Explain each choice a Length of your pencil b Length of a bedroom c Thickness of a piece of wire d Distance from North Bay to Toronto 3 Joanna and Kush want to
18. e value of b then click on the keys 4 2 Click OK From the Measure menu select Calculate different side lengths Click on the value of c then click on the keys A 2 Click OK oe measures ke the one but with Does the calculation of a b and fit the Pythagorean theorem for this triangle Explain 9 Drag a vertex of the triangle to change the values of a and b New Calculation while keeping ZC 90 Does the Pythagorean theorem still hold a b 70 7184 cm 10 Drag point C so that ZC is less than 90 Does the Pythagorean theorem still hold il How do the values of a b and c compare ajej 9 A yae aj 6 6 Erie 11 Drag a vertex of the triangle to change the values of a and b Ft 21S ee while keeping ZC less than 90 J How do the values of a b and c compare eee IL 12 Use the Calculate feature to determine which expression results in the closest value to c for your triangle Cry 2bsnG a Fbt 2abceos C gb 2abtan C 13 Change the size and shape of your triangle while keeping ZC less than 90 Does the same expression give the result closest to c 1 5 The Cosine Law 31 Making a Table B You will need a protractor and a scientific calculator 1 Draw a triangle with three acute angles as large as possible Label the sides and angles of your triangle as shown C b 2 Measure the sides of your triangle ifanallodhanaloy to the nearest
19. eball diamond is a square with side length 90 ft What is the distance between first base and third base Describe the method you used to solve this problem 6 CHAPTER 1 Trigonometry Many fire departments have aerial ladder trucks The ladder extends so that fire fighters can spray water on a fire or rescue people from upper floors Investigate Determining Sine Cosine and Tangent Work with a partner B a You will need a protractor and Make sure your calculator is set to a scientific calculator ad E gt Fach draw a large right triangle mieten SA and label it as shown gt Measure the sides of your triangle to the nearest tenth A C of a centimetre Measure the angles in your triangle to the nearest degree Record the measurements on your triangle gt Copy the table Calculate the ratios for the first 3 rows as decimals Ratios AABC length of side opposite ZA to 2 decimal places length of hypotenuse ar To determine sin A press length of side opposite ZA SIN Then enter the length of side adjacent to ZA measure of ZA and length of side adjacent to ZA press I Record your length of hypotenuse answer in the table to sine of ZA or sinA 2 decimal places Continue cosine of ZA or cos A for cos A and tan A tangent of ZA or tanA How did you determine cos A and tan A Compare your results with other pairs What do you notice about the sine cosine and tangent of
20. egment AB and label it c 1 3 The Sine Law 19 4 5 6 7 9 20 Use the Selection Arrow tool to select side a From the Measure menu select Length The value of a is displayed What is the value of a Selection Deselect the objects Repeat question 4 for sides b and c Use the Selection Arrow tool to select points B A and C in this order From the Measure menu select Angle The measure of ZA will be displayed labelled mZBAC To change the label double click on it using the Text tool Enter A as the new label Then click OK gt The Geometer s Sketchpad Triangle psp fe File Edit Display Construct Transform Measure Graph Window Help a fF g8cm b 45 45cm C 6 47 cm A 7B A 46 Gx 6g Use the Selection Arrow tool to select points A B and C in this order Repeat question 6 for the measure of ZB Select points B C and A in this order Repeat question 6 for the measure of ZC Use the Selection Arrow tool to drag points so that each angle in the triangle is acute What are the side lengths and angle measures of AABC gle measures CHAPTER 1 Trigonometry 10 11 12 13 14 15 16 17 The rati a tan A From the Measure menu select Calculate Click on the value of a click on the a key select tan from the Functions list click on the measure of ZA Click on OK The value of is displayed Follow a outa process
21. ground 10 ft How long should the brace be to the nearest foot 7 Assessment Focus A land surveyor draws a diagram of a plot of land that is in the shape of a right triangle The longest side is 4 7 km long The angle between the longest side and one other side is 59 4 7 km a How long are the two shorter sides b Describe another method to solve the problem E c Which method do you prefer Why 8 A hiker can see a forest ranger tower due north of his position He can see another tower that he knows is 5 0 km due east of the first tower Using his compass he faces the first tower then turns 52 until he is facing the second tower How far is he from each tower 9 Take It Further Jovanna wants to cut a triangular piece off a 4 by 8 sheet of plywood She wants the piece to have one side 4 long and a 30 angle in the corner opposite the 4 edge 8 a How far along the 8 side should she measure to mark the cutting line to the nearest inch Include a diagram b Sketch another way Jovanna could cut the sheet In Your Own Words The cosine of an acute angle in a right triangle has the same measure as the sine of the other acute angle in the triangle Explain why Include a drawing with your explanation 1 1 Sides of Right Triangles 11 Literacy in A Closer Look at Your Text Math Copy and complete the chart Use checkmarks in the left columns to show whether
22. h a triangular top The side lengths 45 cm of the triangle are 45 cm 60 cm and 70 cm 60 cm B To make the triangle Max needs to know 70 cm the angle measures A Since he knows three side lengths he can use the cosine law to find any angle To determine the measure of ZB write the cosine law 2 2 _ h2 cosB Hte re 2ac Substitute a 70 c 45 and b 60 cos B re Press 70 45 cos B 0 5278 60 x2 d2 gt lt 70 x 45 H ZB 58 1 ZB is about 58 1 7 The Sine Law and the Cosine Law 41 1 Determine the unknown measurements from the Investigate 2 Write the equation you would use to determine the cosine of each angle in this triangle P a ZQ b ZR q d c ZP R Q p 3 Decide whether to use the sine law or the cosine law to determine the indicated measure in each triangle Justify your choices a b b ZX 6 4m 5 8m 7 2M l 4 Determine the measure of ZA for those triangles in which you can use the cosine law Explain why you cannot use the cosine law in the other triangle What other strategy could you use to find ZA in that triangle a 10 3 cm B b A c B A 23 0m 9 5 cm 15 A 1 I C 24 2 m C C 5 For each diagram write an equation you can use to determine the unknown measure Then calculate the measure a ZB b ZC c B B 145 yd i C 3 9 cm 4 7 cm a 8 2 km C 127 yd C C 54cm A 13 0 km A B 42 CHAPTER 1 Trigonometry 6 Assessment focus A plane P
23. hapter Review 45 What Should Be Able to Do 1 1 4 Calculate the measure of each unknown side Explain your steps a B C 2 5 cm A D C b S q r R 15m Q 2 A tree casts a shadow 10 m long when the sun s rays make an angle of 25 with the ground a Draw a diagram b What is the height of the tree Which trigonometric ratio did you use 1 2 3 Calculate the measure of the unknown angles Explain your steps xX Y 10 23 46 CHAPTER 1 Trigonometry i 1 3 4 Determine the measures of the unknown ae sides and angles a A 19cm j C a B b C 1 90 cm A a 0 97 cm B 5 Two forest rangers Ren and Micheline observe the same fire from their observation towers Ren s tower is 19 km due east of Micheline s Ren sights the fire on a bearing of 305 from his tower Micheline sights the fire on a bearing of 032 from her tower How far is the fire from each tower N Fire N 32 Micheline 19km a Ren 1 5 6 Determine the measures of the unknown 1 6 sides and angles a B 54 in 97 in 59 in b R 7 7 cm P 8 3 cm Q 7 A bus station is at the intersection of Main St and King St The angle between the streets is 67 Two buses leave the bus station at the same time After 15 min one bus has travelled 2 5 km along Main St The other bus has travelled 5 0 km along King St How far apart are the buses Bus A 2 5 km Main St Station 5 0km BUSE King
24. in Right Triangles 13 Connect the Ideas When you press the calculator displays the measure of the angle whose sine is the number you enter For example if you press 0 75 l the calculator displays 48 59 This shows that sin 48 59 0 75 The and keys work in the same way These function calculator keys represent inverse operations Mira is dropped off 1 4 km due south of the park office The campsite is 3 5 km due east of the park office To walk directly to the campsite in what direction should Mira set her compass Mira s compass is marked at 2 intervals so round the answer to the nearest 2 P is a right angle because the north and east directions intersect at 90 _ Determine the measure of ZH The side opposite ZH is 3 5 km The side adjacent to ZH is 1 4 km Since you know the opposite and adjacent sides use the tangent ratio H length of side opposite ZH aie length of side adjacent to ZH Substitute length of side opposite ZH 3 5 length of side adjacent to ZH 1 4 tan H 22 Press 3 5 l 1 4 DEI ZH 68 2 Mira should set her compass at 68 or walk along a line that makes an angle of about 68 measured clockwise from north 14 CHAPTER 1 Trigonometry Give angle measures to the nearest degree 1 For each triangle determine tan A Then determine the measure of ZA a B b 3 7 cm 47 m po Som A C 2 6 km j 114m C 2 For the given inf
25. l is the tree to the nearest 10 cm 15 0 m 1 1 Sides of Right Triangles 9 Sometimes the unknown measure is the denominator of the ratio not the numerator Example A small plane is flying at an altitude of 3500 m over Lake Ontario toward an island The angle of depression to the island is 15 How much farther to the nearest kilometre does the plane need to fly before it is above the island Solution Let P represent the position of the plane Let q represent the horizontal distance angle of from the plane to the island in metres Sketch a triangle To determine the side length q use the tangent ratio _ length of side opposite to ZP length of side adjacent lt P tan 15 a Multiply each side by q tan 15 xX q 2 X q tan 15 X q 3500 To isolate q divide each side by tan 15 tan 15 Xq _3500 e tan 15 tan 15 _ _3500 C rae cl Press 3500 TAN 15 q 13 062 1 km 1000 a L E i The distance the plane needs to fly before it is above the island is about 13 km 5 In each diagram i Which trigonometric ratio would you use to calculate the length of the labelled side Explain your choice ii Use the ratio to calculate the length to the nearest unit a Z b R S T c T 8 cm 15 ft 23m F YQ 10 CHAPTER 1 Trigonometry 6 A carpenter is cutting a support brace to reach the top of a 10 ft high wall She wants the brace to make an angle of 70 with the
26. l your work for questions 3 to 6 11 ft 3 Knowledge and Understanding Determine the measure of side b to the nearest foot and the measures of ZA and ZC to the nearest degree 4 Communication Explain how you know when to use the sine law or the cosine law in an acute triangle Use an example for each law in your explanation 5 Application A cruise ship sights channel marker A on a bearing of 041 and channel marker B on a bearing of 092 The distance from the ship to channel marker A is 2 0 km The distance from the ship to channel marker B is 2 8 km S 2 8km lB What is the distance between the channel markers l Water Hazard 6 Thinking The layout of a golf hole is shown in the diagram What is the distance from the tee directly to the hole 48 CHAPTER 1 Trigonometry
27. le Investigate f these keystrokes do t work for your Iculator check your s Manual A Reflect Determining Angles in Right Triangles Work with a partner he legs are the two r sides in a right triangle You will need a protractor and a scientific calculator gt Each draw a right triangle in which the ratio of one leg to the other leg is 3 to 5 For example for one triangle draw a right angle with one line segment 6 cm and the other 10 cm Join the endpoints to draw the hypotenuse Label the dimensions And for the other triangle draw a right angle with one segment 9 cm and the other 15 cm gt Measure each acute angle in your triangle Compare measures of corresponding angles in the triangles What do you notice gt Press Enter a tangent ratio for your triangles by pressing 3 5 Press What do you notice gt Press TAN Enter the other tangent ratio for your triangles by pressing 5 3 Press What do you notice gt Repeat this activity using any ratio for the lengths of legs to draw another pair of triangles Are the same relationships true How do you know the ratio of one leg to the other in your first pair of triangles is 3 to 5 Explain how you determined the sides lengths for your second pair of triangles Compare your results with other pairs How does the tangent of an angle appear to be related to the lengths of sides in a right triangle 1 2 Angles
28. m 2 7 cm B 3 3 cm C Solution Use the cosine law formula with a on the left and cos A on the right a amp b 2bc cos A Substitute a 3 3 b 2 7 c 3 8 ber the order of 3 37 2 77 3 8 2 X 2 7 X 3 8 cos A 10 89 21 73 28 08 cos A Subtract 21 73 from each side 10 89 21 73 21 73 20 52 cos A 21 73 10 84 20 52 cos A Divide each side by 20 52 10 84 cos A Press 10 84 20 52 ZA 58 112 ZA is about 58 36 CHAPTER 1 Trigonometry Write the formula you could use to determine the measure of the marked angle in each triangle Then determine the angle measure a ZA b ZB c ZC 4 0 cm 3 9 cm 10 9 m 276 km 213 km 4 2 cm 12 3m 195 km In the Investigate suppose you knew the M of the gorge is 2 5 km but did not know that ZC 73 How could you determine the measure of ZC a Create your own problem about the gorge in the Investigate b Solve your problem c Explain your strategy for creating your own problem d Trade problems with a classmate Solve the problems 8 Assessment Focus A golfer hooks his first drive on a 250 yd golf hole It lands 225 yd from the tee but is 33 off line as shown in the diagram a Determine how far from the hole the ball lands to the nearest yard b The golfer hits his next shot directly toward the hole It travels 121 yd Will the ball reach the hole If so how do you know If not how far fro
29. m the hole will it be to the nearest yard c Explain how you know your solution is correct 1 6 Applying the Cosine Law 37 9 The diagram shows two airplanes approaching a control tower The first plane A is 105 km from the tower T on a bearing of 040 The second plane B is 220 km from the tower T on a bearing of 115 a What is the measure of ZATB b How far apart are the airplanes 10 A ranger sights a forest fire The fire F is 4 8 mi from her observation tower T on a bearing of 304 She calls a forest protection service station S The station is 8 9 mi from the tower on a bearing of 219 a What is the measure of ZFTS b How far is the fire F from the station S 11 Take It Further How are the cosine law and the Pythagorean theorem the same How are they different Include diagrams in your answer In Your Own Words The lengths of two sides in a triangle are known What additional information do you need to use the cosine law Explain two possible solutions Describe two ways you can use this text to find out about the cosine law 38 CHAPTER 1 Trigonometry The Sine Law and the Cosine Law When solving a problem with an acute triangle you may need to decide whether to use the sine law or the cosine law Investigate Choosing the Sine or Cosine Law Work with a partner You will need a scientific calculator Alecia s plan for building a porch shows only some of
30. measurements she needs for the pieces of wood gt Alecia needs the length of b in AABC the measure of ZD in ADEF the length of z in AXYZ and the measure of ZQ in APQR gt Decide whether she should use the sine law or the cosine law to determine each measurement For each triangle record the law you choose entering the known measurements Reflect gt Justify each decision about which law to use gt Is it necessary to solve for each measurement to know whether you chose the law Alecia can use Explain 1 7 The Sine Law and the Cosine Law 39 Connect the Ideas In any AABC Use the sine law when gt you know the measures of two angles and the length of any side gt you know the lengths of two sides and the measure of an angle that is not between these sides Use the cosine law when gt you know the lengths of two sides and the measure of the angle between them b a c 2ac cos B 40 CHAPTER 1 Trigonometry You can solve a b c 2bc cos A for cos A Then determine the measure of an angle a b 2bc cos A Add 2bc cos A to each side a 2bc cos A b 2bc cos A 2bc cos A a 2bc cos A b Subtract a from each side a 2bc cos A g b e a 2bc cos A b e g Divide each side by 2bc 2becosA _ bP e 2bc 2bc Note the position of A ee ae and a in the formula 2bc Max is designing a wooden plant stand C wit
31. open it and begin at question 7 gt Ifyou are not using the file Triangle2 gsp complete all the questions 1 Start the program From the Edit menu select Preferences Set Angle to be measured in degrees to the nearest tenth Set Distance to be measured in centimetres to the nearest hundredth 2 Use the Point tool gt Click on the screen in three places to construct three points Use the Text tool A Click on each point to label the points A B and C 3 Use the Selection Arrow tool to select points A and B Le From the Construct menu select Segment Use the Text tool Double click on line segment BC and label it a Double click on line segment AC and label it b Double click on line segment AB and label it c 30 CHAPTER 1 Trigonometry Use the Selection Use the Selection Arrow tool to select side a R i Deselect the objects Repeat question 5 for sides b and c Use the Selection Arrow tool to select points A C and B Use the Selection Arrow tool to drag point C From the Measure menu select Length The value of a will be displayed in this order From the Measure menu select Angle The measure of ZC will be displayed labelled mZACB To change the label double click on it using the Text tool Enter C as the new label Then click OK so that ZC is as close to 90 as possible From the Measure menu select Calculate Click on the value of a then click on the keys 2 H Click on th
32. ormation in each triangle which angle can be calculated using the sine ratio Determine its measure a Q 17 cm R b X c E ca 11 ft 19 cm 6 ft P 5 3 4m k D F 3 Decide which trigonometric ratio can be used to determine the measure of the marked angle in each diagram Explain each choice Then use the ratio to determine the angle measure a B b T 83m sy 3in A 120m R gt c Z d Z 0 098 km 4 1 cm X 0 025 km X 7 8 cm 1 2 Angles in Right Triangles 15 16 A boat is 100 m from the bottom of a 75 m high cliff Calculate the angle of elevation ZA from the boat to the top of the cliff The Royal Gorge Bridge Incline Railway in Colorado is the steepest railway in the world The elevation increases 1096 ft over the 1550 ft incline 1550 ft 1096 ft Calculate the angle of inclination of the track Sketch each triangle then calculate the angles a In AABC a 1 2 cm b 4 5 cm and ZC 90 Determine the measure of ZB b In ADEF d 43 yd f 92 yd and ZF 90 Determine the measure of ZD c In ANPQ p 49 m q 25 m and ZP 90 Determine the measure of ZN en you sketch you do need to draw to scale A canoeist uses a compass to navigate from N N a lake entry point to a campsite on a small island The campsite is 2 4 km due north and Campsite then 3 2 km due west of the lake entry point eae In what direction should the canoeist head to go directly to the camp
33. s of sides and angles Saima is in a boat directly west of Richard s boat The distance between the boats is 150 ft Each person can see a buoy B from her or his boat Saima sees the buoy on a bearing of 027 Richard sees it on a bearing of 342 How far is Saima s boat from the buoy to the nearest foot Draw a triangle Label its vertices S R and B to represent Saima Richard and the buoy We know the value of b Determine ZB In ASRB the measure of ZS is 90 27 63 West is 3 X 90 270 clockwise from north So subtract 270 to determine the measure The sum of the angles of ZR in ASRB 342 270 72 ina triangle is 180 The measure of ZB is 180 63 72 45 To determine the value of r write the sine law for ASRB with r in the numerator sin 72 sin 63 sin 45 24 CHAPTER 1 Trigonometry O sin 72 sin 45 Multiply each side by sin 72 to isolate r Z X sin 72 0_ x sin 72 s n 72 s n 45 r esnez Press 150 SIN 72 SIN 45 r 201 75 Saima s boat is about 202 ft from the buoy 1 Calculate the value of c in each expression Give your answers to 1 decimal place _ 4 sin 40 losma 6 9 sin 58 a c sin 65 b c sin 75 c c sin 34 2 In Connect the Ideas how far is Richard s boat from the buoy 3 Determine the lengths of the unknown sides in each triangle a X b D c B a
34. se from the north line to a given direction The bearing of B from A is 020 MESTE Using the Sine Law to Solve a Problem Work with a partner You will need a scientific calculator The pilot of an ultralight plane Carmen is flying from Kingston A to Bancroft B Because of an error in compass readings she flies off course by an angle of 25 She travels 50 mi before she realizes that she is going in the wrong direction Her new course is on a bearing of 040 to correct her flight path This triangle represents her situation gt You know that b 50 mi How could you determine the measure of ZB What is its measure Calculate You know ZA 25 What do you know about How can you determine a Calculate this length How far is the plane from Bancroft to the nearest mile when Carmen realizes the error Describe the strategy you used to solve the problem Was it a reasonable strategy Explain How could you determine the distance from Kingston to Bancroft 1 4 Applying the Sine Law 23 Connect the Ideas Thesinelaw N The sine law relates the sides and angles in a triangle In any AABC The length of any side a o a g oe ee a TG divided by the sine of the opposite angle is the same for all 3 pairs of sides and angles The sine of any angle sinA sinB _ sinC a b divided by the length of the opposite side is the same for all 3 pair
35. site Lake entry point Express your answer in degrees clockwise from mn north rounded to the nearest 2 A carpenter is cutting a board that is 14 0 cm wide ra She is using a radial saw that can be set to cut at any angle Po m cut angle To fit under a stairway the board needs to be P eee 17 8 cm shorter on one side than the other ra At what angle should the carpenter make the cut eee to the nearest degree CHAPTER 1 Trigonometry Sometimes you need to determine the length of a side or convert a length to different units before you can use a trigonometric ratio to solve a problem Example Enricho is checking the accuracy of his clinometer A clinometer can be The school s flagpole is 6 0 m tall edoure He walks 5 0 m from the base of the pole le of elevation and holds the clinometer at eye level 120 cm above the ground The clinometer shows the angle of elevation from Enricho s eye to the top of the flagpole as 45 How accurate is the reading on the clinometer to the nearest tenth of a degree Solution Draw a diagram angle of 6 0 m elevation B C 120c 5 0 m The flagpole is perpendicular to the ground and the ground is parallel to the horizontal line of sight So AABC is a right triangle with ZC 90 In AABC the length of the side opposite ZB is 6 0 m 1 2 m 4 8 m The side adjacent to ZB is 5 0 m Use the tangent ratio to calculate the angle of eleva
36. t assumptions did you make Activate Prior Knowledge 5 Pythagorean Theorem oon The Pythagorean theorem relates the area of the square on the hypotenuse to the areas of the squares on the other two sides of a right triangle hypotenuse The Pythagorean theorem states that in a right triangle 2 c a b where c is the length of the hypotenuse and a and b are the lengths of the other two sides Example A builder needs to purchase material for the guy wires of a 56 m tall tower Each guy wire is anchored to the ground 20 m from the base of the tower The material is sold by the metre What is the length needed for each guy wire Solution Sketch a diagram Let the length of a guy wire be c Use c a b Substitute a 20 and b 56 56m f c 202 562 3536 c 3536 c 59 46 The length of material needed for each guy wire is 60 m 60 m instead of 59 m material for each guy wire 1 Determine each unknown length a zit b 20 cm 16 ft 12cm y X 2 Jadie uses an 18 ft ladder to reach the window of the second floor of her house She places the base of the ladder 4 ft 6 in away from the house to set the ladder at a safe angle How far up the wall of the house is the second floor window 3 Corey purchased a 17 flat screen for his computer The height of the screen 11 is 11 Determine its width Would the screen fit into a 14 wide space Explain 4 A bas
37. tenth of a centimetre Length of Measure ZC to the nearest degree i ele Length of b 3 Copy this table and create six blank columns Length of c Record your data in the first four rows ANE of ZC Include data from five classmates in your table b C 4 Calculate the values for the next five rows of the table ah Gp oe st elo Sih a b 2ab cos C 5 The Pythagorean Theorem states that a2 b 2ab tan C if ZC 90 a b c In your triangle ZC is less than 90 Is the Pythagorean Theorem true for your triangle What about for your classmates triangles 6 Look at the other expressions you calculated Which is close to the value of c for your triangle Is this same relationship true for your classmates triangles Reflect The cosine law shows this relationship Cc a b 2ab cos C How does your investigation verify the cosine law Does your investigation show a relationship between c and a b 2ab sin C or between c and a b 2ab tan C Justify your answer Did you use The Geometer s Sketchpad or a table What is an advantage or a disadvantage of this tool 32 CHAPTER 1 Trigonometry Surveyors need to determine distances that cannot be measured directly Investigate A Reflect Using the Cosine Law to Solve a Problem You will need a scientific calculator A surveyor needs to determine the length of a gorge She is standing at a point 2 3 km
38. tion length of side opposite ZB length of side adjacent to ZB Substitute 4 8 and 5 0 tan B Press 4 8 5h ZB 43 8 The clinometer reading was 45 45 43 8 1 2 The device is accurate to within about 1 2 tan B 1 2 Angles in Right Triangles 17 9 A wheelchair ramp is 3 2 m long It rises from ground level to a porch that is 35 cm above the ground For safety reasons there are maximum angles a wheelchair ramp can make with the ground Does this ramp meet either of the following rules Justify each answer Include diagrams a For power wheelchairs the maximum angle is 7 1 b For self propelled wheelchairs the maximum angle is 4 8 10 Assessment Focus A train track runs along a river in a valley The track rises a vertical height of 600 m over a horizontal distance of 25 km a What is the angle of inclination of this section of the track to the nearest tenth of a degree b Create another problem for the diagram Solve your problem c Explain your thinking as you created the problem d Trade problems with a classmate to solve Compare solutions 600 m 25 km 11 Take It Further There are two support wires C on one side of a 15 0 m hydro pole Both wires are attached to the ground 3 5 m from the base of the pole One wire goes to the top of the pole The other wire goes to the middle of the pole M 15 0m What is the angle between the two wires where they meet at the
39. to display a and oa a What do the ratios ae Za Repeat question 11 to display ier ratios and Eo se ac Repeat question 11 to display these ratios What do the ratios compare For which ratios are the values equivalent Drag one or more of the vertices of the triangle to create a new acute triangle while keeping each angle less than 90 Are your observations about equivalent ratios for the first triangle still true Repeat this process at least five more times New Calculation 7 8 9 Values d aj sjej APPENEE o Arcsin Help Cancel Arccos Arctan The ratio sa compares the sine of ZA to the value of the side opposite ZA What do the ratios ne and me compare Have the program calculate and display these ratios What do you notice Drag one or more of the vertices of the triangle to create a new acute triangle Are your observations from question 16 still true Making a Table You will need a protractor and a scientific calculator 1 Draw a large triangle with three acute angles Label the sides and angles of your triangle as shown 1 3 The Sine Law 21 2 Measure the sides of your triangle to the nearest tenth of a centimetre Measure the angles of your triangle to the nearest degree 3 Copy this table and create six blank columns EAA Record your data in the first six rows of one column
40. you agree or disagree with each statement C ws mmm O O oe O gt Disagree Agree Disagree Math texts are mainly questions teachers assign for homework Math texts can be read just like any other book use a math text only when instructed to do so by my teacher e Compare your thoughts on these statements with a partner Discuss reasons for any differences With your partner complete this Text Quest Look at the title page in a few chapters List the features of the title pages This textbook has two indices What are they Why is each index useful Identify three types of technology addressed in the book Each chapter has a Literacy in Math feature Identify three topics covered in these features that you think will be useful Look at Connect the Ideas in a few lessons What is the purpose of the trapezoids In the Mid Chapter Review and What Should I Be Able to Do there are red numbers in the margin What do these numbers represent Choose something else in the text that you think can help you Explain how it can help 12 e Complete the right columns of your chart e Explain whether your thinking has changed Why or why not CHAPTER 1 Trigonometry Mira is hiking due south of a park office She knows there is a campsite due east of the park office To hike directly to the campsite she needs to determine the direction to set her compass This requires determining an angle in a right triang

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