Ch1 e-textbook
Contents
1. What You ll Learn To determine the measures of sides and angles in right acute and obtuse triangles and to solve related problems And Why Applications of trigonometry arise in land surveying navigation cartography computer graphics machining medical imaging and meteorology where problems call for calculations involving angles lengths and distances using indirect measurements lt lt lt ae Activate Prior Knowledge The Pythagorean Theorem Prior Knowledge for 1 1 The hypotenuse of a right triangle is the side opposite the right angle It is the longest side A Pythagorean Theorem In right AABC with hypotenuse c p F Cn aby C 3 B Determine the unknown length b C Materials b 15 ft A l A scientific calculator Solution 21 ft B Use the Pythagorean Theorem in AABC c b Subs tute c 2l anda 15 A 56 ab ye Subtract 15 from both sides to isolate b Write your answer QE 52 e Take the square root of both sides to isolate b to the same number nal places i NDE Press 21 B 15 as the least accurate 14 70 g emen eed So side b is about 15 feet long in calculations CHECK EZ 1 Determine each unknown length a 250 km b 17 45 m 60 aan Z 23 27 m 4 2 In isosceles right APQR ZP 90 PR 6 7 m p FR F Determine the length of QR aj 3 What is the length of the diagonal across the soccer field 2 CHAPTER 1 Trigonometry Metr2. 10 11 12 Is each trigonometric ratio positive or negative Explain how you know a tan 53 b cos 96 sin 132 Is 2B acute or obtuse Explain a tan B 1 6 c cos B 0 35 b cos B 0 9945 d sin B 0 7 a cos A 0 94 Determine cos 180 A b sin A 0 52 Determine sin 180 A c tan A 0 37 Determine tan 180 A Determine the measure of each obtuse angle a sin R 0 93 b cos D 0 56 0 lt ZG lt 180 Is ZG acute or obtuse How do you know a tan G 0 2125 b sin G 0 087 Determine the measure of ZA If you get more than one answer for the measure of Z A explain why a tan A 0 1746 b sin A 0 3584 ZZ is an angle in a triangle Determine all possible values for Z Z a cos Z 0 93 b sin Z 0 73 13 Use the Sine Law to determine x and y y 32 68 17mm 17 One side of a triangular lot is 2 6 m long EH 18 14 a Determine the lengths of sides p and q b Explain your strategy 15 Solve each triangle Sketch a diagram first a ALMN ZM 48 ZN 105 andl 17m b AHI ZH 21 ZJ 57 and h 9 feet 4 inches 16 different triangular sections of a sailboat sail Lani received these specifications for two 19 20 a a 5 5 m b 1 0 m and ZC 134 Determine ZA b d 7 75 m e 9 25 m and ZF 45 Determine ZE B 21 The angles in the tria
3. Determining the Measure of an Angle Using the Sine Law a Calculate the measure of ZZ in AXYZ b Determine the unknown side lengths x and y Solution a ZZ 180 83 35 62 So ZZ is 62 b Write the length of side z in inches 2 feet 2 inches 26 inches Write the Sine Law for AXYZ sin X sin Y sin Z To find x use the first and the third ratios Le Substitute 7X 83 ZZ 62 and z 26 sin X sin Z x 26 2 e Multiply each side by sin 83 to isolate x 26 o A nT x s n 83 29 25 So side x is about 30 inches or about 2 feet 6 inches long To find y use the second and third ratios in the Sine Law for AXYZ a ae Substitute LY 35 ZZ 62 and z 26 rs maan Multiply each side by sin 35 to isolate y y x sin 35 16 89 So side y is about 17 inches or about 1 foot 5 inches long 1 4 The Sine Law 29 Example 3 Applying the Sine Law Materials A plane is approaching a 7500 m runway e scientific calculator The angles of depression to the ends of the runway are 9 and 16 How far is the plane from each end of the runway Solution To determine ZR in APQR use the angles of depression ZPRQ 16 9 7 Write the Sine Law for APQR a ee sin P sin Q sin R To determine q use the second and third ratios d r 1 o o inQ smR Substitute ZR 7 ZQ 164
4. and r 7500 Se Multiply each side by sin 164 to isolate sin 164 sin 7 y y d 7500 gme sin 164 16963 09 ZP 180 164 7 9 To determine p use the first and third ratios in the Sine Law for APQR P f sin P sin R Substitute ZR 7 ZP 9 and r 7500 P _ 7500 T 8 Multiply each side by sin 9 to isolate p p a x sin 9 9627 18 So the plane is about 16 963 m from one end and about 9627 m from the other end of the runway 30 CHAPTER 1 Trigonometry 1 Write the Sine Law for each triangle Circle the ratios you would use to m For help with calculate each indicated length questions 1 and 2 a X b y tor c N Example 1 see Example seca 23m Y 3 0 ft M N p X T Z 2 For each triangle in question 1 calculate each indicated length 3 Use the Sine Law to determine the length of each indicated length a E b K A A G F j E For help with 4 a Write the Sine Law for each triangle questions 4 and 5 b Circle the ratios you would use to find each unknown side see Example 2 E K G 120 22m 69 2 4 km M 76 n F L Z Y 39 13 cm X X 121 14 2 mi Z Y 1 4 The Sine Law 31 c Solve each triangle i X il lil 5 Solve each triangle a X b 32 6 Solve each triangle a X 128 Z 4cm 1 ft 2 in 8 mm 11 Y 7 Choose one part from question 6 Write to explain how you solved the triang
5. the plot AB a Solution C Ten tan A ng a in Substitute BC 1 8 and AC 1 2 42 11 8 LA tan 28 2nd TAN 1 8 1 2 D ERIE 56 31 The measure of ZA is about 56 ZB 180 ZC ZA 180 90 56 34 The measure of ZB is about 34 b ZBAC and Z TAB form a straight line So ZTAB 180 56 124 The bearing of the third side of the plot is about 124 C 1 8 km The sum of the angles in riangle is 180 6 CHAPTER 1 Trigonometry Example 3 Materials e scientific calculator An angle of elevation is Solving Problems Jenny and Nathan want to determine the height of the Pickering wind turbine Jenny stands 60 0 feet from the base of the turbine She measures the angle of elevation to the top of the turbine to be 81 On the other side of the turbine Nathan measures an angle of elevation of 76 Jenny and Nathan each hold their clinometers about 4 8 feet above the ground when measuring the angle of elevation a What is the height of the wind turbine b How far away from the base is Nathan z A Solution Sketch and label a diagram Drawing a diagram may help you visualize the information in the problem Your diagram does not need to be drawn to scale For example this X 4 6 ft diagram is not drawn to 60 0 ft scale a Use AAKJ to find the length of AK tan J a Substitute AK h ZJ 81 and JK 60 0 tan 81 ae M
6. A Is tan A positive or negative Explain why 9 Suppose ZA is obtuse Use your answers from question 7 a Think about sin A A Is sin A positive or negative Explain why b Think about cos A Is cos A positive or negative Explain why c Think about tan A A Is tan A positive or negative Explain why 18 CHAPTER 1 Trigonometry iim 1 Use your work from Inquire For each angle measure is point P in Quadrant I or Quadrant II a 35 b 127 c 95 2 Is the sine of each angle positive or negative a 45 b 67 c 153 3 Is the cosine of each angle positive or negative a 168 b 32 c 114 4 Is the tangent of each angle positive or negative a 123 b 22 c 102 5 Is each ratio positive or negative Justify your answers a tan 40 b cos 120 c tan 150 d sin 101 e cos 98 f sin 13 6 Is ZA acute obtuse or either Justify your answers a cos A 0 35 b tan A 0 72 c sin A 0 99 m What did your results show about the measure of ZA and its trigonometric ratios when point P is in Quadrant I m What did your results show about the measure of ZA and its trigonometric ratios when point P is in Quadrant II 1 2 Investigating the Sine Cosine and Tangent of Obtuse Angles 19 Surveyors and navigators often work with angles greater than 90 They need to know how to interpret their calculations Investigate Exploring Supplem
7. cos 4 Ea 4 Substitute a 3 5 b 5 2 c 5 6 150 oo 52 S 2x 3 5X 5 6 64 99 So the measure of ZB is about 65 ZA 180 77 65 38 So the measure of ZA is about 38 1 6 Problem Solving with Oblique Triangles 45 Example 3 Materials e scientific calculator To ensure accurate results when calculating b use the value for a before rounding Solving Two Oblique Triangles Determine lengths a and b Solution In oblique AABC we know the measures of two angles and the length of the side opposite one of them AAS So use the Sine Law Write the Sine Law for AABC sin A sin B sin C To find a use the first and the third ratios a Substitute ZC 77 ZA 54 and c 6 sin A sin C _ 6 E T A Multiply both sides by sin 54 to isolate a 6 o a zz X sin 54 4 98 So a is about 5 m long In oblique ABCD we now know the lengths of two sides and the measure of the angle between them SAS So use the Cosine Law Write the Cosine Law for ABCD using ZB b a 2accosB Substitute a 4 98 c 8 ZB 93 4 987 8 2x 4 98 x 8 x cos 93 92 97 b V92 97 9 64 So b is about 10 m long 46 CHAPTER 1 Trigonometry 1 a E For help with questions 1 and 2 see Example 1 2 3 E For help with 4 questions 4 and 5 see Example 2 5 6 Decide whether you would use the Sine Law or the Cosine Law to de
8. hoe C C The Sine Law lt Oe e are one OG a In any AABC sinA sinB sin C B B To use the Sine Law you must know at least one side angle pair for example a and ZA To determine a side length you must know an additional angle measure e angle angle side AAS e angle side angle ASA The Cosine Law In any AABC a b 2ab cos C To determine a side length using the Cosine Law you must know e side angle side SAS To determine an angle measure using the Cosine Law you must know e side side side SSS Study Guide 53 54 41 Determine each indicated measure a B b X 1 7 m a Fas mm Z L b Y 445 mm 2 Danny is building a ski jump with an angle of elevation of 15 and a ramp length of 4 5 m How high will the ski jump be 3 Determine the measure of Z CBD A R 7 3 in 5 2 in 4 A triangular lot is located at the intersection of two perpendicular streets The lot extends 350 feet along one street and 450 feet along the other street a What angle does the third side of the lot make with each road b What is the perimeter of the lot Explain your strategy 350 ft 450 ft CHAPTER 1 Trigonometry Chapter Review i 5 A ship navigator knows that an island harbour is 20 km north and 35 km west of the ship s current position On what bearing could the ship sail directly to the harbour Answer questions 6 to 8 without using a calculator 6
9. 15 3107 cos 52 147 0001 t V 147 0001 12 12 The planes are about 12 1 miles apart 1 5 The Cosine Law 37 1 Write the Cosine Law you would use to determine each indicated side E For help with questions 1 11 ft 2 in and 2 see 14 Z NU m e Example 1 ae N 2 Determine each unknown side length in question 1 3 Determine each unknown side eosin a M 2 7 ft 1 3cm O 3 0 ft 2 4 cm 4 Determine the length of a A b a C 10 1 cm B a 15 9 cm A a E For help with 5 Write the Cosine Law you would use to determine the measure of the questions 5 and 6 marked angle in each triangle see Example 2 a B 3 2 cm b p 145 yd 111 yd 5 0 cm 4 3 cm R 35 yd 4 11 km 6 23 km M978 km 6 Determine the measure of each marked angle in question 5 7 Determine the measure of each unknown angle Y 112 mm 18 8 cm 21 8 cm 118 i 545 mm 540 mm 14 9 cm 14 9 cm a 38 CHAPTER 1 Trigonometry E For help with 10 questions 10 and 11 see Example 3 11 Determine the measure of each unknown angle Start by sketching the triangle a In AXYZ z 18 8 cm x 24 8 cm and y 9 8 cm Determine ZX b In AGHJ g 23 feet h 25 feet and j 31 feet Determine Z J c In AKLM k 12 9 yards 17 8 yards m 14 8 yards Determine ZM Would you use the Cosine Law or the Sine Law to determine the labelled length in each triangle Explain your reasoning a A
10. 45 The cosine of an obtuse angle is 0 45 Calculate the sine of this angle to 4 decimal places The sine of an obtuse angle is Calculate the cosine of this angle to 4 decimal places a Determine the values of sin 90 and cos 90 b Explain why tan 90 is undefined In Your Own Words Is knowing a trigonometric ratio of an angle enough to determine the measure of the angle If so explain why If not what else do you need to know CHAPTER 1 Trigonometry Materials grid paper Scissors protractor scientific calculator Rules for drawing AABC e Draw all vertices on intersecting grid lines e Do not draw any vertex on an axis Home eri Trigonometric Search a Play with a partner m Each player e Cuts out two 20 by 12 grids Draws and labels each grid as shown Draws AABC on the first grid Writes the x and y coordinates of each vertex e Joins vertices A B and C with the origin O Measures the angle made by each vertex with the positive x axis e Calculates a trigonometric ratio of their choice for the angle measured In turn states the x or y coordinate and the calculated trigonometric ratio for the angle measured m The other player e Uses the ratio to calculate the angle made with the positive x axis e Uses the x or y coordinate to plot the point e Uses primary trigonometric ratios to find the second coordin
11. Sine Law 33 i s A 3 R Collecting Important Ideas A Focusing on key ideas can help you improve performance at both 5 ill work and school As you prepare for college or a career keep track of the important ideas you learn 2 ee When recording important ideas m Use captions arrows and colours to help show the idea m Add a picture a diagram or an example m Keep it brief Don t say more than you need to Here s an example The Sine Law B Bn aS sin A sin B sinC sinA _ ginB _ sinC a b C You need one pair and one more side or angle A h C 1 What changes would you make to the example above 2 Begin with this chapter Make a collection of trigonometry ideas Create a section in your math notebook or start a file on your computer 3 During the year add to your collection by creating a record of important ideas from later chapters 34 Transitions Collecting Important Ideas A furniture designer s work begins with a concept developed on paper or ona computer then built to test its practicality and functionality The designs may then be mass produced Precise angle measurements are required at all times Investigate Using Cosine Ratios in Triangles Materials e protractor e scientific calculator AABC a Right triangle Acute triangle Obtuse triangle Work with a partner m Construct a AABC for each description e LC 90 All angles
12. V Xi Xi i i Compare strategies with another pair How are they different How are they the same How did you decide which law or laws to use Justify your choices Could you have used another law Explain why or why not 42 CHAPTER 1 Trigonometry Connect the Ideas Sine Law Cosine Law Solving triangle problems In any triangle we can use the Sine Law to solve the triangle when m We know the measures of two angles and the length of any side AAS or ASA In any triangle we can use the Cosine Law to solve the triangle when m We know the lengths of two sides and the measure of the angle between them SAS m We know the lengths of three sides SSS When solving triangle problems m Read the problem carefully m Sketch a diagram if one is not given Record known measurements on your diagram m Identify the unknown and known measures m Use a triangle relationship to determine the unknown measures Often two or more steps may be needed to solve a triangle problem Sketch a diagram The sum of angles in any Right AABC Oblique AABC triangle is 180 The Pythagorean Primary Sine Law Cosine Law Theorem trigonometric ratios 1 6 Problem Solving with Oblique Triangles 43 Example 1 Materials e scientific calculator need to use the Sine Law or the Cosine Law to solve it Determining Side Lengths Decide whether to use the Sine Law or Cosine Law to determine m and p in A
13. a Is tan A positive or negative Explain why d Move point P around in Quadrant I What do the sine cosine and tangent calculations show about your work for parts a b and c 9 Suppose ZA is obtuse Use your answers from question 7 a Think about sin A A Is sin A positive or negative Explain why b Think about cos A Is cos A positive or negative Explain why c Think about tan A a Is tan A positive or negative Explain why d Move point P around in Quadrant II What do the sine cosine and tangent calculations on screen show about your work for parts a b and c 16 CHAPTER 1 Trigonometry ae length of side opposite ZA length of hypotenuse ae length of side adjacent to 2A length of hypotenuse length of side opposite 7 A E length of side adjacent to ZA Using Pencil and Paper Work with a partner Part A Investigating Trigonometric Ratios Using Point P x y 1 On grid paper draw a point P x y in Quadrant I of a coordinate grid Label the sides and vertices as shown 2 In right APBA how can you use the Pythagorean Theorem and the values of x and y to determine the number of units for side r 3 In right APBA a What is the measure of 7 A b Which side is opposite 7 A Adjacent to 7 A Which side is the hypotenuse c Use x y and r Write each ratio i sin A ii cos A iii tan A 4 On the same grid choose a second posit
14. b What is the angle measure between the shorter sides Explain your strategies Use this diagram of the rafters in a greenhouse a What angle do the rafters form at the peak of the greenhouse b What angle do they form with the sides of the greenhouse Solve this problem two ways using the Cosine Law and using primary trigonometric ratios 26 ft 6 in 48 CHAPTER 1 Trigonometry 12 13 14 15 ere 16 A boat sails from Meaford to Christian Island to Collingwood Christian Island then to Wasaga Beach a What is the total distance h i mi s _ Wasaga the boat sailed rr e Beach b What is the shortest distance from Wasaga Beach to Christian Island A triangle has side lengths measuring 5 inches 10 inches and 7 inches Determine the angles in the triangle Assessment Focus Roof rafters and truss form oblique APQR and ASQT a Describe two different methods that could be used to determine Z SQT b Determine Z SQT using one of the methods described in part a c Explain why you chose the method you used in part b Literacy in Math Explain the flow chart in Connect the Ideas in your own words Add any other important information Two boats leave port at the same time One sails at 30 km h on a bearing of 305 The other sails at 27 km h on a bearing of 333 How far apart are the boats after 2 hours In Your Own Words What mistakes can someone make w
15. s Sketchpad e TechSinCosTan gsp e scientific calculator Work with a partner The intersection of the x axis Bi f reates four Part A Investigating Trigonometric Ratios Using Point P x y regions called quadrants m Open the file TechSinCosTan gsp qe numbered counterclockwise Make sure your screen looks like this starting from the upper right P 12 36 5 98 The angle made by line segment r with the positive x axis is labelled 2A 1 2 Investigating the Sine Cosine and Tangent of Obtuse Angles 13 1 Use the Selection Arrow tool k pad Click on Show Triangle PBA then deselect the triangle 2 Move point P x y around in Quadrant I In right APBA how can you use the Pythagorean Theorem and the values of x and y to determine the length of side r Click on Show r Compare your method with the formula on the screen 3 Choose a position for point P in Quadrant I In right APBA a What is the measure of 7 A b Which side is opposite Z A Adjacent to ZA Which side is the hypotenuse c Use x y and r Write each ratio cts za i sin A ii cos A iii tan A th of hypotenuse Click on Show xyr Ratios of side adjacent to ZA Compare your answers with the ratios on the screen ngth of hypotenuse h of side opposite 2A side adjacent to ZA 14 CHAPTER 1 Trigonometry 4 Choose a position for point P in
16. that is not a right triangle The Sine Law The Sine Law relates the sides and the angles in any oblique triangle acute triangle is an M ange The Sine Law tuse triangle is an In any oblique AABC 7 A B 2 oblique triangle Geo i eg 8 b a sinA sinB sinC r b C sinA _ sinB _ sinC Acute AABC Obtuse AABC a 7 C Angle Angle Side When we know the measures of two angles in a triangle and the length AAS of a side opposite one of the angles we can use the Sine Law to determine the length of the side opposite the other angle Example 1 Determining the Length of a Side Using the Sine Law Materials What is the length of side d in ADEF e scientific calculator E d 95 f 38 cm 40 D Solution Write the Sine Law for ADEF d _ e f sinD sinE sinE Use the two ratios that include the known measures d e Substitute 7D 40 ZE 95 and e 38 sin D sin E d _ 38 l OE a Multiply each side by sin 40 to isolate d d pss X sin 40 38 E E 950 amp E 40 D E 24 52 So side d is about 25 cm long 28 CHAPTER 1 Trigonometry Angle Side Angle ASA Example 2 Materials e scientific calculator 1 fot 12 inches When we know the measure of two angles in a triangle and the length of the side between them we can determine the measure of the unknown angle using the sum of the angles in a triangle then we can use the Sine Law to solve the triangle
17. A A b A c B 1 3 cm A C 322 0 m c 4 5 ft 2 5 cm C 4 5 ft B B 6 Use trigonometric ratios and the Pythagorean Theorem to solve each a triangle a B b P c 13 6 ft 7 2 ft 31 0 cm 33 in A C r b R D 13 6 cm Q 7 A ladder 10 feet long is leaning against 2 ft 7 in a wall at a 71 angle a How far from the wall is the foot of the ladder a b How high up the wall does the ladder reach 8 The Skylon Tower in Niagara Falls is about 160 m high From a certain distance Frankie measures the angle of elevation to the top of the tower to be 65 Then he walks another 20 m away from the tower in the same direction and measures the angle of elevation again Use primary trigonometric ratios to determine the measure of the new angle of elevation 1 1 Trigonometric Ratios in Right Triangles 9 E For help with 9 A rescue helicopter is flying horizontally at an altitude of 1500 feet over question 9 see Example 3 10 Georgian Bay toward Beausoleil Island The angle of depression to the island is 9 How much farther must the helicopter fly before it is above the island Give your answer to the nearest mile ee as pean a 1500 ft A theatre lighting technician adjusts the light to fall on the stage 3 5 m away from a point directly below the lighting fixture The technician measures the angle of elevation from the lighted point on the stage to the fixture to be 56 What is the height
18. MNP Then determine each side length Solution In oblique AMNP we know the measures of two angles and the length of the side between them ASA So use the Sine Law Write the Sine Law for AMNP m _ n P sin M sin N sin P ZN 180 15 23 142 To find m use the first two ratios ae Substitute n 33 ZM 15 and ZN 142 sin M sin N m 33 ae es ee Multiply both sides by sin 15 to isolate m 33 o n a n 15 m 13 87 m is about 14 yards To find p use the second and third ratios TE Substitute n 33 ZN 142 and ZP 23 oe aes Multiply both sides by sin 23 to isolate p _ 33 ee a p 20 94 p is about 21 yards 44 CHAPTER 1 Trigonometry ZELIA Determining Angle Measures Materials Write the equation you would use to determine the cosine of each angle e scientific calculator in AABC Determine each angle measure A 5 2 m 5 6 m C 3 5 m B Solution In oblique AABC we are given the lengths of all three sides SSS So use the Cosine Law Write the Cosine Law using ZC c a b 2ab cos C Isolate cos C 2 2_ 2 cos C 1 Isolate C 2 2 2 C cos 4 a c Substitute a 3 5 b 5 2 c 5 6 T Ra 56 po 2x3 5X5 2 77 AD So the measure of ZC is about 77 Write the Cosine Law using ZB b a2 2ac cos B Isolate cos B 2 2 12 cos B ete Isolate B oi oe i B
19. P sin P ZP sin 0 65 0 65 ZP 40 5 Using a calculator 180 ZP 180 40 5 sin 0 65 40 5 139 5 180 40 5 139 5 b cos R 0 22 b cos 180 R cosR ZR cos 0 22 22 ZR 77 3 Using a calculator 180 ZR 180 77 3 cos 0 22 102 7 102 7 c tan S 0 44 c tan 180 S tan S ZS tan 0 44 44 ZS 23 7 Using a calculator 180 ZS 180 23 7 tan 0 44 23 7 156 3 180 23 7 156 3 22 CHAPTER 1 Trigonometry E For help with question 1 see Example 1 E For help with questions 4 5 and 6 see Example 2 Each point on this coordinate Determine the sine cosine and tangent ratios for each angle Give each answer to 4 decimal places a 110 b 154 c 102 Is each trigonometric ratio positive or negative Use a calculator to check your answer a sin 35 b tan 154 c cos 134 grid makes a right triangle with the origin A and the x axis Determine the indicated trigonometric ratio in each triangle a sin A and point P b cos A and point Q c tan A and point R d cos A and point S e sin A and point T f tan A and point V Suppose ZP is an obtuse angle Determine the measure of 7 P for each sine ratio Give each angle measure to the nearest degree a 0 23 b 0 98 c 0 57 d 0 09 Determi
20. Quadrant II a What is the measure of 7 A b In right APBA what is the measure of Z PAB c Is the x coordinate of P positive or negative d Is the y coordinate of P positive or negative 5 Copy the table Use a scientific calculator for parts a and b Angle measure sinA cosA tan A Acute ZA Obtuse ZA An acute angle is less than l 90 An obtuse angle is a Use the measure of ZA from question 3 Complete the row for between 90 and 180 acute Z A b Use the measure of ZA from question 4 Complete the row for obtuse ZA c Click on Show Ratio Calculations Compare with the results in the table Part B Determining Signs of Trigonometric Ratios m Use the Selection Arrow tool Move point P around Quadrants I and II 6 a Which type of angle is 7 A if point P is in Quadrant I b Which type of angle is ZA if point P is in Quadrant II 7 a Can r Vx y be negative Why or why not b When ZA is acute is x positive or negative c When ZA is obtuse is x positive or negative d When ZA is acute is y positive or negative e When ZA is obtuse is y positive or negative 1 2 Investigating the Sine Cosine and Tangent of Obtuse Angles 15 8 Suppose ZA is acute Use your answers from question 7 a Think about sin A A Is sin A positive or negative Explain why b Think about cos A Is cos A positive or negative Explain why c Think about tan A
21. al or both How important is the accuracy of measurements and calculations What types of communication are used written graphical or both What other mathematics does this occupation involve What is a typical wage for an entry level position in this career What is a typical wage for someone with experience in this occupation Are employees paid on an hourly or a salary basis m Read about the occupation you chose A H A on the Internet or in printed materials m If possible interview people who work in the occupation you chose Prepare a list of questions you would ask them about trigonometry Search words occupational information working conditions other qualifications earnings salaries related occupations oe fo 0 O 1 7 Occupations Using Trigonometry 51 MaN ee ANAN SMAN AMAN ANA SS S S S S Search words o training education information O apprenticeship programs Part C Researching Educational Requirements m Research to find out the educational requirements for the occupation you chose On the Internet use search words related to education You might decide to use the same words you used before Find out about any pre apprenticeship training programs or apprenticeship programs available locally Go to Web sites of community colleges or other post secondary institutions e Use course calendars of post secondary institutions or ot
22. are acute e ZC is obtuse E Copy and complete the table b h c c cosC 2abcosC a b 2ab cos C m What patterns do you notice in the table m Compare your results with other pairs Are your results true for all triangles The Cosine Law is c a b 2ab cos C How are the Cosine Law for triangles and the Pythagorean Theorem the same How are they different How does the information in your table show this 1 5 The Cosine Law 35 Connect the Ideas The Cosine Law Side Angle Side SAS Example 1 Materials e scientific calculator The Sine Law cannot be ere because no ratios includes the ly the order of operations e given measures mM p The Sine Law although helpful has limited applications We cannot use the Sine Law to solve an oblique triangle unless we know the measures of at least two angles We need another method when m We know the lengths of two sides and the measure of the angle between them SAS m We know all three side lengths in a triangle SSS The Cosine Law can be used in both of these situations The Cosine Law In any oblique AABC c a b 2ab cos C C E 3 Acute triangle Obtuse triangle In any triangle given the lengths of two sides and the measure of the angle between them we can use the Cosine Law to determine the length of the third side Determining a Side Length Using the Cosine Law Determine the length of n in AMNP S
23. ate of the vertex e Marks the findings on the second grid Player A Player B sin A 0 16 ZA ain 016 9 and x units y o tono What are the coordinates of vertex A oy tano xo 095 20 AlO T The player who first solves the other player s triangle wins the round Repeat the game with different triangles m What strategy did you use to determine the other coordinate of each vertex m What are the most common mistakes one can make during the game How can you avoid them GAME Trigonometric Search 25 Mid Chapter Review 1 Solve each triangle a B C 32 cm A dC b b D 13 yd C 27 yd 2 Solve each right AXYZ Sketch a diagram a LX 53 ZZ 90 x 3 6 cm b z 3 feet 5 inches x 25 inches ZZ 90 3 A flight of stairs has steps that are 14 inches deep and 12 inches high A handrail runs along the wall in line with the steps What is the angle of elevation of the handrail 12 in 4 Cables stretch from each end of a bridge to a 4 5 m column on the bridge The angles of elevation from each end of the bridge to the top of the column are 10 and 14 What is the length of the bridge 45m 10 14 gt lt 26 CHAPTER 1 Trigonometry 5 John hikes 2 5 miles due west from Temagami fire tower then 6 miles due north N 6 mi l a How far is he from the fire tower at the end of the hike b What beari
24. b c 39 75 km A C 14 6 m a C C B 15 9 m B A harbour master uses a radar to monitor two ships B and C as they approach the harbour H One ship is 5 3 miles from the harbour on a bearing of 032 The other ship is 7 4 miles away from the harbour on a bearing of 295 a How are the bearings shown in the diagram b How far apart are the two ships A theatre set builder s plans show a triangular set with two sides that measure 3 feet 6 inches and 4 feet 9 inches The angle between these sides is 45 Determine the length of the third side 1 5 The Cosine Law 39 12 A telescoping ladder has a pair of aluminum struts called ladder stabilizers and a base What is the angle between the base and the ladder 2 1m 1 1m 13 A hydro pole needs two guy wires for support What angle does each wire make with the ground 14 A land survey shows that a triangular plot of land has side lengths 2 5 miles 3 5 miles and 1 5 miles Determine the angles in the triangle Explain how this problem could be done in more than one way 15 Assessment Focus An aircraft navigator knows that town A is 71 km due north of the airport town B is 201 km from the airport and towns A and B are 241 km apart a On what bearing should she plan the course from the airport to town B Include a diagram b Explain how you solved the problem T p lt r ae 40 CHAPTER 1 Trigonometry 16 17 18 Literacy in Mat
25. d you use the results of your calculations from parts a and b Justify your answer 4 Tell what you like about your stage Why is your stage appropriate for a concert or theatre What would make someone choose your design for a stage 58 CHAPTER 1 Trigonometry
26. e each trigonometric ratio for C to 4 decimal places a sin C b cos C c tan C Solution Use a calculator a sin C sin 123 b cos C cos 123 c tan C tan 123 0 8387 5446 1 5399 1 3 Sine Cosine and Tangent of Obtuse Angles 21 Supplementary angles sum of the measures of Example 2 Materials e scientific calculator since different angles have trigonometric your calculator may n the measure of e angle when you inverse trigonometric armine the measure of P 180 sin sin A cos cos A ZA 180 tan tan A See how these properties are yoplied in Method 2 In Lesson 1 2 we investigated relationships between trigonometric ratios of an acute angle and its supplement We can use these relationships to determine the measure of an obtuse angle Properties of supplementary angles Given an acute angle A and its supplementary obtuse angle 180 A e sin A sin 180 A e cos A cos 180 A tan A tan 180 A Determining the Measure of an Obtuse Angle Write the measure of each supplementary obtuse angle when a The sine of acute ZP is 0 65 b The cosine of acute ZR is 0 22 c The tangent of acute ZS is 0 44 Solution Method 1 First determine the measure of the acute angle Method 2 First determine the trigonometric ratio of the obtuse angle a sin P 0 65 a sin 180
27. entary Angles Materials Work with a partner e scientific calculator m Copy and complete the table for LA 25 The sum of the measures ZA 25 Supplementary angle of two supplementary angles is 180 sin A cos A tan A m Repeat for 7B 105 and for ZC 150 m Examine your tables for 7A 7B and ZC What relationships do you see between e The sines of supplementary angles e The cosines of supplementary angles e The tangents of supplementary angles 20 CHAPTER 1 Trigonometry xyr definition When ZA is obtuse x is legative Signs of the primary trigonometric ratios Materials e scientific calculator m If you know the trigonometric ratios of an acute angle how can you determine the ratios of its supplementary angle E If you know the trigonometric ratios of an obtuse angle how can you determine the ratios of its supplementary angle Connect the Ideas The trigonometric ratios can be defined using a point P x y on a coordinate grid Trigonometric ratios In right APBA sin A 2 r 0 cosA r 0 tan A Z x 0 When point P is in Quadrant I When point P is in Quadrant II ZA is acute Z A is obtuse m sin A is positive m sin A is positive m cos A is positive m cos A is negative m tan A is positive m tan A is negative Determining Trigonometric Ratios of an Obtuse Angle Suppose ZC 123 Determin
28. gth of side adjacentto ZA b b m We can use the primary trigonometric ratios or combinations of these ratios to determine unknown measures Determining Side Lengths Determine the length of p in AMNP M p 225 fi NE P Solution In AMNP e The length of the hypotenuse is given e The measure of acute Z P is given pis opposite ZP So use the sine ratio sin P MN Substitute MN p ZP 60 and MP 225 MP sin 60 ae Multiply both sides by 225 to isolate p sin 60 x 225 p ress SIN 60 225 p 194 86 The length of p is about 195 feet 1 1 Trigonometric Ratios in Right Triangles 5 1 Inverse ratios m You can use the inverse ratios sin t cos and tan to determine the measure of an angle when its trigonometric ratio is known the key strokes shown Press or to access the inverse ratios on do not work on your jlator refer to the manual a scientific calculator Inverse ratios A For acute ZA in right AABC sin sin A A cos cos A A tan tan A A m In navigation and land surveying direction is described using a bearing The bearing is given as a three digit angle between 000 and 360 measured clockwise from the north line Example 2 Determining Angle Measures Materials Michelle is drawing a map of a triangular N e scientific calculator plot of land a Determine the angles in the triangle b Determine the bearing of the third side of
29. h Write a problem that can be represented by this diagram Solve your problem Ashley 175m Bill d 225 m Cheryl Use this diagram of a roof truss 41 ft a Determine the length of TR b What is the angle of elevation Record your answer to the nearest degree Marie wants to determine the height of an Internet transmission tower Due to several obstructions she has to use indirect measurements to determine the tower height She walks 50 m from the base of the tower turns 110 and then walks another 75 m Then she measures the angle of elevation to the top of the tower to be 25 50 m d a What is the height of the tower b What assumptions did you make Do you think it is reasonable to make these assumptions Justify your answer In Your Own Words How can the Pythagorean Theorem help you remember the Cosine Law Include a diagram with your explanation 1 5 The Cosine Law 41 Campfires a part of many camping experiences can be dangerous if proper precautions are not taken Forest rangers advise campers to light their fires in designated locations away from trees tents or other fire hazards Decide whether to use the Sine Law or the Cosine Law to determine each distance Choosing Sine Law or Cosine Law Work with a partner Two forest rangers sight a campfire F from their observation towers G and H How far is the fire from each observation deck 4 0 mi y y V
30. hen to use the Sine Law or the Cosine Law to solve a triangle problem Give examples to illustrate your explanation 5 Application A car windshield wiper is 22 inches long Through which angle did the blade in this diagram rotate 6 Thinking A sailboat leaves Port Hope and sails 23 km due east then 34 km due south Port Hope A ei O a On what bearing will the boat travel on its way back to the starting point 34 km b How far is the boat from the starting point _ lt gt c What assumptions did you make to answer 1 l 4 parts a and b Practice Test 57 Designing a Stage You have been asked to oversee the design and construction of a concert and theatre stage in your local community park TEZ aT fen 22 3S y Pis AV VA ONO ne 1 1 Design a stage using at least two right triangles and at least two oblique triangles Make decisions about the stage e What will be the shape of the stage e Will it have a roof e Where will the stairs be What will they look like Will you include a ramp e Include any further details you consider important for your stage 2 Estimate reasonable angle measures and side lengths Vhat tools did you use Mark them on your drawing 3 a Describe two calculations for designing your stage that include a right triangle Show your calculations b Describe two calculations for designing your stage that include an oblique triangle Show your calculations c How coul
31. her information available through your school s guidance department m How can you get financial help for the required studies How could you find out more Part D Presenting Your Findings m Prepare a presentation you might give your class another group or someone you know Think about how you can organize and clearly present the information and data you researched Use diagrams or graphic organizers if they help m Write about a problem you might encounter in the occupation you researched What would you do to solve it How does your solution involve trigonometry m Why do you think it is important to have a good understanding of trigonometry in the occupation you researched 52 CHAPTER 1 Trigonometry Study Guide Primary Trigonometric Ratios When ZA is an acute angle in a right AABC i length of side opposite LA q sin A length of hypotenuse C length of side adjacent to ZA cosA e length of hypotenuse e length of side opposite 7 A nh e lt length of side adjacent to ZA b o and ana The inverse ratios are sin e sin sin A A ecos cos A lt tan nA A Trigonometric Ratios of Supplementary Angles Two angles are supplementary if their sum is 180 For an acute angle A and its supplementary obtuse angle 180 A e sin A sin 180 A e cos A cos 180 A A e tan A tan 180 A
32. her 75 What is the height of the tree What assumptions did you make Determine the length x of the lean to roof attached to the side of the cabin CHAPTER 1 Trigonometry 26 27 28 29 30 Two ferries leave dock B at the same time One travels 2900 m on a bearing of 098 The other travels 2450 m on a bearing of 132 a How are the bearings shown on the diagram b How far apart are the ferries a Write a word problem that can be solved using the Sine Law Explain your strategy b Write a word problem that can be solved with the Cosine Law Explain your strategy Describe one occupation that uses trigonometry Give a specific example of a calculation a person in this occupation might perform as part of their work Tell about an apprenticeship program or post secondary institution that offers the training required by a person in the career you identified in question 29 Practice Test Multiple Choice Choose the correct answer for questions 1 and 2 Justify each choice 1 Which could you use to determine the measure of an angle in an oblique triangle if you only know the lengths of all three sides A Sine Law B Cosine Law c Tangent ratio D Sine ratio 2 If ZA is obtuse which is positive A sin A B cos A C tan A D none Show all your work for questions 3 to 6 3 Knowledge and Understanding Solve AABC A C 5 9m 4 Communication How do you decide w
33. hile solving problems with the Sine Law and the Cosine Law How can they be avoided 1 6 Problem Solving with Oblique Triangles 49 Po 1 7 ro Professional tools such as tilt indicators on boom trucks lasers and global positioning systems GPS perform trigonometric computations automatically CITT Researching Applications of Trigonometry Materials It may be helpful to invite an expert from a field that uses trigonometry e computers with Internet or an advisor from a college or apprenticeship program access Work in small groups Part A Planning the Research m Brainstorm a list of occupations that involve the use of trigonometry Include occupations you read about during this chapter m Briefly describe how each occupation you listed involves the use of trigonometry Use these questions to guide you e What measures and calculations might each occupation require e What tools and technologies might each occupation use for indirect measurements How do these tools and technologies work m Find sources about career guidance Write some information that might be of interest to you or to someone you know 50 CHAPTER 1 Trigonometry Part B Gathering Information m Choose one occupation from your list Investigate as many different applications of trigonometry as you can in the occupation you chose Include answers to questions such as What measurement system is used metric imperi
34. ic and Imperial Unit Conversions Prior Knowledge for 1 1 a The metric system is based on powers of 10 Metric conversions l cm 10 mm linn 100 cm 1 km 1000 m 1 mm 0 1 cm 1 cm 0 01 m 1 m 0 001 km The most common imperial units of length are the inch foot yard and mile Imperial conversions 1 foot 12 inches 1 yard 3 feet 1 mile 5280 feet l yard 36 inches 1 mile 1760 yards Write each pair of measures using the same unit a 54 cm 3 8 m b 22 inches 12 feet 4 inches Solution a 1 m 100 cm so 3 8 m 3 8 x 100 380 cm Alternatively 1 cm 0 01 m so 54 cm 54 x 0 01 0 54 m b 1 foot 12 inches so 12 feet 144 inches 12 feet 4 inches 144 inches 4 inches 148 inches Alternatively 22 inches 1 foot 10 inches CHECK 4 Convert each metric measure to the unit indicated a 7 2 cm to millimetres b 9215 m to kilometres c 9 35 km to metres d 832 cm to metres e 879 m to centimetres f 65 mm to metres 2 Convert each imperial measure to the unit indicated a 7 feet to inches b 28 yards to feet c 8 miles to feet d 963 feet to yards e 23 feet 5 inches to inches f 48 inches to feet 3 Determine q If you need to convert measurements to a different unit explain why a b 223 mm q 8 ft 0 45 m 3 yd 1 ft Activate Prior Knowledge 3 Specialists in forestry and arboriculture apply trigonometry to determine heights of trees They may use a clinometer an instrument f
35. ion for point P in Quadrant II Label as shown Re aai FENHA y ING ENHE Ael NYa B ixl JA00 a What is the measure of 7 A b In right APBA what is the measure of Z PAB c Is the x coordinate of P positive or negative d Is the y coordinate of P positive or negative 1 2 Investigating the Sine Cosine and Tangent of Obtuse Angles 17 5 Copy the table Angle measure sinA cosA tan A Acute ZA Obtuse ZA Use a scientific calculator a Use the measure of 7 A and the number of units for each side from question 3 Complete the row for acute ZA b Use the measure of ZA and the number of units for each side from question 4 Complete the row for obtuse ZA Part B Determining Signs of the Trigonometric Ratios eis less than 6 a Which type of angle is ZA if point P is in Quadrant I An obtuse angle is b Which type of angle is ZA if point P is in Quadrant II between 90 and 180 7 a Can r Vx y be negative Why or why not b When ZA is acute is x positive or negative c When ZA is obtuse is x positive or negative d When ZA is acute is y positive or negative e When ZA is obtuse is y positive or negative 8 Suppose ZA is acute Use your answers from question 7 a Think about sin A A Is sin A positive or negative Explain why b Think about cos A Is cos A positive or negative Explain why c Think about tan A
36. le 8 Could you use the Sine Law to determine the length of side a in each triangle If not explain why not a A b A c B 5 4 cm 9 ft 8 ft C 20 a 23 m a B B a C cu A 9 In question 8 determine a where possible using the Sine Law 10 a What is the measure of ZT in ATUV b Determine the length of side a U 3 a 15 yd ce 88 T V 11 a Use the Sine Law to determine the length of side b b Explain why you would use the Sine Law to solve this problem c Which pair of ratios did you use to solve for b Explain your choice A 14 2 cm CHAPTER 1 Trigonometry E For help with question 15 see Example 3 12 13 14 16 In AXYZ a Determine the lengths of sides x and z b Explain your strategy In ADEF ZD 29 ZE 113 and f 4 2 inches a Determine the length of side e b Explain how you solved the problem Literacy in Math Describe the steps for solving question 13 Draw a diagram to show your work A welder needs to cut this triangular shape from a piece of metal R Q P Determine the measure of ZQ and the side lengths PQ and QR Assessment Focus A surveyor is mapping a triangular plot of land Determine the unknown side lengths and angle measure in the triangle Describe your strategy In Your Own Words What information do you need in a triangle to be able to use the Sine Law How do you decide which pair of ratios to use 1 4 The
37. ne the measure of ZR for each cosine ratio Give each angle measure to the nearest degree a 0 67 b 0 56 0 23 d 0 25 Determine the measure of 7 Q for each tangent ratio Give each angle measure to the nearest degree a 0 46 b 1 60 c 0 70 d 1 53 ZM is between 0 and 180 Is ZM acute or obtuse How do you know a cos M 0 6 b cos M 0 6 c sin M 0 6 For each trigonometric ratio identify whether ZY could be between 0 and 180 Justify your answer a cos Y 0 83 b sin Y 0 11 c tan Y 0 57 d tan Y 0 97 1 3 Sine Cosine and Tangent of Obtuse Angles 23 24 9 10 11 12 13 14 15 Literacy in Math Create a table to organize information from this lesson about the sine cosine and tangent ratios of supplementary angles Use Connect the Ideas to help you Acute angle Obtuse angle sine positive positive Cosine Z Gis an angle in a triangle Determine all measures of ZG a sin G 0 62 b cos G 0 85 c tan G 0 21 d tan G 0 32 e cos G 0 71 f sin G 0 77 The measure of ZY is between 0 and 180 Which equations result in two different values for LY How do you know a sin Y 0 32 b sin Y 0 23 c cos Y 0 45 d cos Y 0 38 e tan Y 0 70 f tan Y 0 77 Assessment Focus Determine all measures of ZA in a triangle given each ratio Explain your thinking a sin A 0 45 b cos A 0 45 c tan A 0
38. ng should he use to return 2 5 mi to the fire tower Answer questions 6 and 7 without using a calculator 6 Determine whether the sine cosine and tangent of the angle is positive or negative Explain how you know a 27 b 95 c 138 7 Is ZP acute or obtuse Explain a cos P 0 46 b tan P 1 43 c sin P 0 5 d cos P 0 5877 8 Determine each measure of obtuse ZP a sin P 0 22 b cos P 0 98 c tan P 1 57 d sin P 0 37 9 ZG is an angle in a triangle Determine all possible values for ZG a sin G 0 53 b cos G 0 42 c tan G 0 14 d sin G 0 05 The Sine Law The team of computer drafters working on the Michael Lee Chin Crystal at the Royal Ontario Museum needed to know the precise measures of angles and sides in triangles that were not right triangles Investigate Relating Sine Ratios in Triangles Work with a partner Draw two large triangles one acute and one obtuse Label the vertices of each triangle A B and C Copy and complete the table for each triangle Angle Angle Sine of Length of Ratios measure angle opposite side ZA a a zis ZB b sinB a A0 C mae as Describe any relationships you notice in the tables Compare your results with other pairs Are the relationships true for all triangles How does the Investigate support this 1 4 The Sine Law 27 Connect the Ideas An oblique triangle is any triangle
39. ngle at each end of the 2 6 m side are 38 and 94 Determine the lengths of the other two sides of the lot a Explain why you would use the Cosine Law to determine 4 R Q 5 0m P b Which version of the Cosine Law would you use to solve this problem Explain how you know the version you chose is correct c Determine the length of side q Sketch and label ABCD with b 7 5 km d 4 3 km and ZG 131 Solve ABCD Determine the measure of Z Q Q 255 9 m 378 6 m R 444 5m a Sketch each triangle b Determine the measure of the specified angle i In AMNO m 3 6 m n 10 7 m and o 730 cm Determine ZN ii In ACDE c 66 feet d 52 feet and e 59 feet Determine ZD 59 Chapter Review 22 Jane is drawing an orienteering map that shows the location of three campsites Determine the two missing angle measures A Campsite 1 A Campsite 3 3 4 km 4 8 km A Campsite 2 23 A machinist is cutting out a large 24 25 56 triangular piece of metal to make a part for a crane The sides of the piece measure 4 feet 10 inches 3 feet 10 inches and 5 feet 2 inches What are the angles between the sides Ren e and Andi volunteered to help scientists measure the heights of trees in old growth forests in Algonquin Park The two volunteers are 20 m apart on opposite sides of an aspen The angle of elevation from one volunteer to the top of the tree is 65 and from the ot
40. of the lighting fixture f a 3 5 m 10 CHAPTER 1 Trigonometry Assume Kenya measures the angle of elevation from the ground to the top of the tower measures an measures an 11 12 13 Kenya s class is having a contest to find the tallest building in Ottawa Kenya chose the Place de Ville tower Standing 28 5 m from the base of the tower she measured an angle of elevation of 72 to its top Use Kenya s measurements to determine the height of the tower A ship s chief navigator is z a Second plotting the course for a tour S of three islands The first island is 12 miles due west of the second island The third island 18 mi is 18 miles due south of the W second island a Do you have enough eT i information to determine island the bearing required to sail directly back from the third island to the first island Explain b If your answer to part a is yes describe how the navigator would determine the bearing Assessment Focus A carpenter is building a bookshelf against the sloped ceiling of an attic a Determine the length of the sloped ceiling AB Be used to build the bookshelf b Determine the measure of Z A sloped ceiling Is Z A an angle of inclination or 35m an angle of depression Why c Describe another method to solve part b Which method do you prefer Why A 3 24 m C 1 1 Trigonometric Ratios in Right Triangles 11 12 14 A
41. olution Write the Cosine Law for AMNP n m p 2mp cos N Substitute m 13 p 25 and ZN 105 13 25 2 x 13 x 25 x cos 105 Press 13 25 x2 962 2323 2 fx 13 B25 n 962 2323 cos 105 D EE 31 02 n is about 31 cm long 36 CHAPTER 1 Trigonometry Side Side Side SSS Example 2 Materials e scientific calculator Example 3 Materials e scientific calculator The number of decimal places in your answer uld match the given measures We can use the Cosine Law to calculate the measure of an angle in a triangle when the lengths of all three sides are known Determine an Angle Measure Using the Cosine Law Determine the measure of ZC in ABCD C 500 ft 650 ft B 750 ft D Solution Write the Cosine Law for ABCD using ZC c b d 2bd cos C Substitute c 750 b 650 d 500 750 6507 5007 2 x 650 x 500 x cos C 562 500 422 500 250 000 650 000 x cos C 110 000 650 000 x cosC Isolate cos C _ _ 110000 ORG na Isolate C cos 10 000 ei 0 650 Ta ZC 80 2569 ZC is about 80 Navigating Using the Cosine Law An air traffic controller at T is tracking two planes U and V flying at the same altitude How far apart are the planes Solution ZLUTV 75 23 52 Write the Cosine Law for ATUV using ZT u v 2uvcos T Substitute u 15 3 v 10 7 ZT 52 15 5 4 10 77 2 X
42. or measuring angles of elevation Choosing Trigonometric Ratios Work with a partner An arborist uses a clinometer to determine the height of a tree during a hazard evaluation This diagram shows the arborist s measurements Use AABC g Determine the lengths of BC and AC For accuracy keep more Use AACD decimal places in your calculations than you Determine the length of CD need in the final answer What is the height of the tree Describe the strategies you used to determine the height of the tree What angles and trigonometric ratios did you use Compare your results and strategies with another pair How are they similar How are they different 4 CHAPTER 1 Trigonometry Connect the Ideas Primary trigonometric ratios 1 vertex is labelled with pital letter Each side is labelled with the lowercase of the opposite vertex Example 1 Materials e scientific calculator ite the length of p to arest foot because ength of n is to the rest foot The word trigonometry means measurement of a triangle m The primary trigonometric ratios are sine cosine and tangent The primary trigonometric ratios B For acute ZA in right AABC oe length of side opposite ZA _ gq ae length of hypotenuse Hypotenuse Side opposite ZA C a length of side adjacent to ZA cosA length of hypotenuse o length of side opposite ZA a A C tan Side adjacent to ZA 7 len
43. roof has the shape of an isosceles triangle 15 16 een 17 angle of inclination 5 9m a What is the measure of the angle of inclination of the roof b What is the measure of the angle marked in red c Write your own problem about the roof Make sure you can use the primary trigonometric ratios to solve it Solve your problem Literacy in Math In right AABC with ZC 90 sin A cos B Explain why Two boats F and G sail to the harbour H Boat F sails 3 2 km on a bearing of 176 Boat G sails 2 5 km on a bearing of 145 Determine the distance from each boat straight to the shore Use paper and a ruler Draw a right AABC where a sin A cos B 0 5 b tan A tan B 1 0 Describe your method In Your Own Words Explain why someone might need to use primary trigonometric ratios in daily life or a future career CHAPTER 1 Trigonometry Investigating the Sine Cosine and Tangent of Obtuse Angles To create a proper joint between two pieces of wood a carpenter needs to measure the angle between them When corners meet at a right angle the process is relatively simple When pieces of wood are joined a at an acute or an obtuse angle the task of creating a proper joint is more difficult CITE Exploring Trigonometric Ratios Materials Choose Using The Geometer s Sketchpad or Using Pencil and Paper e The Geometer s Sketchpad or grid paper and protractor Using The Geometer
44. termine each indicated length a H b 23 c X Q 23 In p 8 mi e a G KK N p R 12 32 h y 38 Z Y 3 2 km Determine each indicated length in question 1 Use the Sine Law or the Cosine Law to determine each indicated length a y b K c 132 D 10 in t 1 ft 1 in 3 2 mi m 9 3 km C 93 24 B d T V U L M 10 4 km Decide whether you would use the Sine Law or the Cosine Law to determine each angle measure b Q 25cm R c E 10 50 cm 4 0 km 2 3 km Determine the measure of each angle in question 4 Use the Sine Law or the Cosine Law to determine each measure for ZB a By b A 3 ft 5 in 3 ft 3 in c C g 1ft 1 6 Problem Solving with Oblique Triangles 47 E For help with 7 questions 7 and 8 see Example 3 E 1 0 sum of the angles in a quadrilateral is 360 11 Phoebe and Holden are on T opposite sides of a tall tree gt 125 m apart The angles of i elevation from each to the top yore of the tree are 47 and 36 What is the height of the tree n I 125 m Carrie says she can use the Cosine Law X to solve AXYZ Do you agree 240 yd Justify your answer Y 206 yd Use what you know from question 8 Write your own question that can be solved using the Sine Law or the Cosine Law Show your solution A hobby craft designer is designing this two dimensional kite a What is the angle measure between the longer sides
45. ultiply both sides by 60 0 to isolate h h tan 81 x 60 0 gt a aie h 378 83 The turbine is about 378 8 feet above eye level 378 8 feet 4 8 feet 383 6 feet The height of the wind turbine is about 384 feet b In right AAKN the measure of ZN and the length of the opposite side AK are known tan N Substitute KN q ZN 76 and AK 378 8 tan 76 aS Rearrange the equation to isolate q 378 8 4 tan 76 ress 378 8 TAN 76 O EXER q 94 4 Nathan is about 94 feet away from the base of the turbine 1 1 Trigonometric Ratios in Right Triangles 7 1 For each triangle name each side in two different ways a Hypotenuse b Side opposite the marked angle c Side adjacent to the marked angle i R 2 Write each trigonometric ratio as a ratio of sides a sin A b cosA c cos B d tan B WA A E For help with 3 Which primary trigonometric ratio can you use to calculate the length of questions 3 and each indicated side 4 see Example 1 a b 103 ft P 7 3 4m 5 q p M E OD c F g H 43 in Q G 4 Use the ratios you found in question 3 to calculate the length of each indicated side 8 CHAPTER 1 Trigonometry E For help with question 5 see Example 2 Give each angle book to recall any measurement conversions E For help with questions 8 and 9 see Example 3 5 For each triangle determine tan A Then determine the measure of Z
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