Find the length, x, in the diagram, rounded to the nearest tenth of a centimetre.

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1 The tangent ratio relates two sides of a right triangle and an angle. If ou know an angle and the length of one of the legs of the triangle, ou can find the length of the other leg. Eample Find a Side Length Using the Tangent Ratio Find the length,, in the diagram, rounded to the nearest tenth of a centimetre. Solution 9.2 cm 28 Write the tangent ratio for the given angle in terms of the side lengths. tan 28 tan (tan 28 ) Multipl both sides b *28 t= or 9.2 *t28 = The length of side is about.9 cm. Eample 5 Solve a Multi-Step Problem Using the Tangent Ratio A radio transmitter is to be supported with a gu wire, as shown. The wire is to form a angle with the ground and reach 30 m up the transmitter. The wire can be ordered in whole-number lengths of metres. How much wire should be ordered? Solution gu wire 30 m Use the tangent ratio of the given angle to find the distance, d, from the tower that the gu wire should be secured. Then, appl the Pthagorean theorem to find the length, w, of wire needed. tan tan 30 d d(tan ) 30 Multipl both sides b d. 30 d Divide both sides b tan.!tan! w d 30 m 360 MHR hapter 7

2 Appl the Pthagorean theorem to find the length of wire needed. w 2 d w 2 30 a 900 tan! b w a 30 tan! b w The length of gu wire needed is about 33.1 m. It can onl be ordered in whole-number lengths. 33 m is too short, so round up to the net metre. At least 3 m of gu wire should be ordered. Ke oncepts The ratio of the side to the side of an angle in a right triangle is called the tangent of that angle. tan You can use a scientific or graphing calculator to epress the tangent of an angle as a decimal θ find one of the acute angles when both leg lengths are known in a right triangle find a side length if one acute angle and one leg of a right triangle are known ommunicate Your Understanding 1 Eplain how the tangent of the slope angle is related to the slope of a ramp. 2 3 a) Eplain how ou can identif the and sides of a right triangle. b) Eplain how these can change in a given triangle. Use a diagram to support our eplanation. a) Eplain what each calculator function does: tangent inverse tangent b) When would ou use each function? c) How are these functions related to each other? 7.3 The Tangent Ratio MHR 361

3 Practise For help with questions 1 and 2, see Eample Find the tangent of the angle indicated, to four decimal places. a) b) c) A d) 2 m 5 m e) D f) 6 m F θ 6 mm mm 2. Refer to question 1. Find the tangent of the other acute angle, to four decimal places. For help with question 3, see Eample Evaluate with a calculator. Record our answer to four decimal places. a) tan b) tan 15 c) tan 62 d) tan 5 e) tan 30.7 f) tan 82. g) tan 20.5 h) tan 5 For help with questions to 8, see Eample 3.. Find the measure of each angle, to the nearest degree. a) tan 1.5 b) tan A c) tan d) tan W e) tan f) tan g) tan X h) tan F m E D 3.5 cm 3 cm 15 mm.8 cm A 17 mm 3 cm E θ 5. Find the measures of both acute angles in each triangle, to the nearest degree. a) 11 mm b) A m c) R d) P 6. Find the length of the unknown side, to the nearest tenth. a) b).2 cm 58 c) d) 8.7 mm 7. Find the length of, to the nearest tenth of a metre. a) b) cm 70 c) 12 m d) 3 8 cm e) f) 63 m k 5 cm Q 1 M A 15 p m 9 m L K cm m 12 m 2 m 362 MHR hapter 7

4 8. Find the length of, to the nearest tenth of a metre. a) b) m 11. Rocco and iff are two koalas sitting at the top of two eucalptus trees, which are located apart, as shown. Rocco s tree is eactl half as tall as iff s tree. From Rocco s point of view, the angle separating iff and the base of his tree is onnect and Appl 9. To measure the width of a river, Kirstn uses a large rock, an oak tree, and an elm tree, which are positioned as shown. rock 70 How high off the ground is each koala? 12. Police are responding to a distress call: elm Show how Kirstn can use the tangent ratio to find the width of the river, to the nearest metre. 10. A surveor is positioned at a traffic intersection, viewing a marker on the other side of the street. The marker is 1 from the intersection. The surveor cannot measure the width directl because there is too much traffic. Find the width of James Street, to the nearest tenth of a metre m James Street M 6 oak S Help needed at 228 Sunset oulevard immediatel. Police headquarters and the trouble site are shown..0 km 2.5 km 228 Sunset oulevard Police Headquarters 590 hestnut Street Squad cars and a helicopter are both immediatel dispatched to the site from headquarters. a) At what angle to hestnut Street should the helicopter travel? b) Assuming that the squad cars can travel at an average speed of 60 km/h and the helicopter can travel twice as fast, how much longer will it take for the squad cars to reach the site than the helicopter? c) Describe an assumptions ou make in our solutions. 7.3 The Tangent Ratio MHR 363

5 13. The diagram shows the roof of a house. How wide is the house, to the nearest metre? rafter 26 3 m 1. Petra walked diagonall across a rectangular schoolard measuring 5 m b 65 m. To the nearest degree, at what angle with respect to the shorter side did she walk? omfortable stairs have a slope of. What angle do the stairs make with the horizontal, to the nearest degree? 16. Find the length of, then the length of, to the nearest tenth of a metre Find the length of, to the nearest tenth of a centimetre, then the measure of, to the nearest degree. 18. To measure the height of a building, hico notes that its shadow is 8.5 m long. He also finds that a line joining the top of the building to the tip of the shadow forms a angle with the flat ground. a) Draw a diagram to illustrate this situation. b) Find the height of the building, to the nearest tenth of a metre. 19. a) Find the tangent of several angles with values between 1 and. Organize our results in a table like this one. Angle, rafter 26 tan b) What is the value of tan 5? 8 cm 9 cm 8 c) Add the tangents of several angles with values between 6 and 89 to our table. d) Add the tangents of several angles with values ver close to, but not equal to, 90 to our table. e) ased on our findings, what conclusions can ou make about the tangents of angles less than 5? greater than 5 and less than 90? ver close to, but not equal to, 90? f) Use the definition of the tangent ratio and geometric reasoning to justif our conclusions. Include diagrams in our eplanation. 20. a) Use a calculator to evaluate the following: tan 0 tan 90 b) Use the definition of the tangent ratio and geometric reasoning to eplain our results. Include diagrams in our eplanation. 21. At hocke practice, Lars has the puck in front of the net, as shown.? He is eactl awa from the middle of the net, which is 2 m wide. Within what angle must Lars fire his shot in order to get it in the net, to the nearest degree? 22. Refer to question 21. a) Does Lars have a better chance, a worse chance, or the same chance to score if he positions himself directl in front of one of the posts, as shown? Eplain our reasoning and an assumptions ou make.? b) Repeat part a) for the case in which Lars moves directl closer to the net directl farther from the net 36 MHR hapter 7

6 Etend 23. a) Make a table of values for tan for values of between 0 and 90. b) Graph the relationship. Is the relationship linear or non-linear? Eplain. c) Describe the shape of the graph and an interesting features ou can identif. 2. The angle at which a skier slides down a hill with a coefficient of friction,, at a constant speed, is given b tan. Natalie is skiing on a hill with a reported coefficient of friction of 0.6. If Natalie skis down at a constant speed, what is the angle of the hill? 25. The tangent ratio is used to design the bank angle for a curved section of roadwa. 26. For each graph, i) find the slope of the line ii) draw a triangle to find the tangent of the acute angle that the line makes with the -ais iii) compare our answers to parts i) and ii) iv) find the acute angle in part ii) c a b 8 10 Let be the bank angle required for a speed limit, v, in kilometres per hour, and a radius, r, in metres. The angle and the speed limit are related b the formula v 2 tan. Find the bank angle required 9.8r for a highwa curve of radius 50 m that will carr traffic moving at 100 km/h. Did You Know? This same relation applies to banking a biccle or motorccle when going around a curve, and banking an aircraft in a turn. 27. Math ontest How man numbers less than contain at least one 5? A D 625 E Math ontest In the figure, PR PQ and RPS 30. IF PS PT, what is the measure of QST? P T R S Q 7.3 The Tangent Ratio MHR 365

7 7.3 The Tangent Ratio, pages a) b) c) d) e) f) a) b) c) d) e) f) a) 2.15 b) c) d) e) f) 7.97 g) h) a) 56 b) 37 c) 31 d) 39 e) 0 f) 1 g) 72 h) a) A 36, 5 b) M 51, K 39 c) P 32, R 58 d) A 53, a) 6.7 cm b) 1.0 m c) 10.0 mm d).1 cm 7. a) 11.0 cm b) 6.0 m c) 11.2 m d) 11.3 m e) 5.1 m f) 13.1 m 8. a) 5.2 m b) 23.6 m 9. tan 72 w ; width is 37 m Rocco s height above the ground: 7 m; iff s height above the ground: 1 m 12. a) 32 b).1 min c) Answers will var m m, 6.3 m cm, a) Answers ma var. b) 18.2 m 8.5 m 19. a) Tables will var. b) tan 5 1 c) Answers will var. d) Answers will var. e) Tangents of angles less than 5 are between 0 and 1; tangents of angles greater than 5 and less than 90 are greater than 1; tangents of angles ver close to, but not equal to, 90 are ver large, and approach infinit. f) Answers ma var. For eample: When the angle is less than 5, the side is shorter than the side, so the tangent ratio is less than 1. When the angle is greater than 5 but less than 90, the side is longer than the side, so the tangent ratio is greater than 1. When the angle gets ver close to 90, the side gets ver small compared to the side, so their quotient becomes ver large. 20. a) tan 0 0; tan 90 is undefined. b) Answers ma var. For eample: When the angle is 0, the side length is zero, so zero divided b an length equals 0. When the angle is 90, the side length is zero, and an side length divided b 0 is undefined a) Answers ma var. For eample: He has a slightl larger angle (1.25 compared to 1.0 ) being positioned in the middle, but normall a plaer slaps the puck predominantl in one direction, so positioning in front of a post might be better. b) Answers ma var. For eample: If he is directl closer to the net, he has a wider angle to work with, so this would be easier. If he is directl farther from the net, he has less of an angle to work with, so this would be more difficult. 23. a) Tables will var. b) Answers will var = tan The relationship is non-linear because the graph is not a straight line. c) Answers will var. The graph looks like it increases ver quickl as it approaches a) i) m 1 ii) tan A 1 iii) The answers are the same. iv) A 5 b) i) m 2 ii) tan 2 iii) The answers are the same. iv) 63 c) i) m 0.5 ii) tan 0.5 iii) The answers are the same. iv) The Sine and osine Ratios, pages a) sin b) sin c) sin d) sin e) sin f) sin g) sin h) sin Answers MHR 557