Solving Right. Triangle Problems. Grant is a seaplane pilot in British Columbia. When he lands, he

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1 .. IuJee1 9 So/vnq Rjh T? 9fr Pro 243 B. How else might a pilot use trigonometry during a flight? A. How can knowing about angles and trigonometry help Grant land his plane? needs to know exactly where the plane will touch down. Grant is a seaplane pilot in British Columbia. When he lands, he Triangle Problems Solving Right F

2 244 App,ericesri. anc i/orkplace 11 be 30. If it is not, try 0.5 I! Sin S = 0.5. To determine LS, enter 4 ijn z. Your answer shoud Make sure your calculator is in Degree mode. I p,., or?n.!ri. keys to determine the angle measure. If you know the sine, cosine, or tangent of an angle, you can use the Determining Unknown Angles L 3. Calculate. b) cosa=,or e) cosb=,or needed for most of calculator (trig this chapter) a scientific a)4xsin25 = You will need functions are length. Label each length, to one decimal place. C) tan A =, or f) tan B =,or for LA are trigonometric ratios The three d) sin B =, or. 3 c A () 14.0 cm 25.0 in. 9.0 in. a) (14.0 cm) 22.0 cm 2 + (22.0 cm) 1. Use the Pythagorean theorem to calculate the unknown side 9 Getting 2. What is each trigonometric ratio for the triangle B b () adjacent opposite tan A hypotenuse adjacent cos A sina opposite hypotenuse sine of LA. a) sina =,or sinameans Hint on the right, as a fraction and as a decimal? 3 1 cm2 + hypotenuy. The Pythagorean Hint cos 56 b) = 2 = c2 (9.0 in.) 2 + b2 = (25.0 in.) cm2 = b2 = (25.0 in.) cm c In. a cm2 = c2 b2 = in

3 ZF sirr 1 means the sine. a) sin C = b) tanf= inverse of the 4. What is the measure of each angle, to the nearest degree? Hint 3 : TrrigIe so LA = LD c 370 AB B CD, In this diagram, 23 b):<.r a) A equal. interior angles are lines, the alternate crosses parallel When a transversal 47) Mark the angle measurements on the diagrams. 6. What are the unknown angle measures in each triangle? to one decimal place. SoPQ= cm, Hint Q PQ= cm 35 A PQxcos35 = cos 35 = //t\\\ b) side PQ cos 35 = hypotenuse adjacent to one decimal place. So CA = cm, =CA X sin 55 = CA sin 55 CA sin 55 = hypotenuse a) side CA opposite on the diagram. 5. How long is each side, to one decimal place? Label the length 1 = tan siw1(o.8387) = /7

4 246 1 e cesp arc /J-<pac i 3houd she use? Express your answer to 4 decal places. Whch gauge-bock height e!evaton must be exactly For this project, the angle of Diock to the rafter. measure angles very accurately. gauge a sine bar and gauge block to should Arya cut the ridge beam? Shaina s a machirst. She uses Example I to use a sine, \ problem? ratio to solve the cosine, or tangent decide whether How did you ) What equation can you use to calculate x? elevation of the rafter, or x. At what angle ridge beam F : If cut j angle of elevation? a ruler You will need rafters 0 Which is the measure of the Angles of Elevation -i(i. /n- V., 6ft Arya should cut at an angle of about What is the cut angle, to the nearest degree? x = XE1 Angle of elevation = The cut angle must match the angle of 12ft cut the ridge beam at an angle to attach it angle company in Medicine Hat. He needs to Arya builds roof trusses for a construction Angle of elevation = 29cm 33cm ii) Write an expression to use for calculating the angle of elevation.

5 h =r Na Chapter 9 Solving Right Triangle Problems 247 The ramp length should be -, to two decimal places. Explain, two decimal places. must not be + = Does this ramp 1 meet the code? steeper than D. Use the Pythagorean theorem. Calculate the ramp length, to wheelchair ramp the slope of a building code, According to the C. Is your answer reasonable? Explain. The angle of elevation should be about XL_J B. What is the ramp s angle of elevation, to the nearest degree? right triangle 900 angle in a that form the Label the lengths of the legs. Use x to represent the angle legs of elevation and r to represent the ramp length. the two sides the base of the porch must be 4.25 m. Solution The ramp must meet the porch 0.35 m above the ground. Vanessa needs to build a wheelchair ramp for a cottage in Whiteshell. Example 2 The distance along the ground from the start of the ramp to What angle of elevation and ramp length should Vanessa use? B. The height of the gauge block should be A. Sketch a right triangle to show a side view of the ramp. in. A. What is an equation for the gauge-block height, h? Solution

6 248 Apprenticeship and Workplace 11 0 Question 2? the side engths n determine one of strategy to a d1ferent x z,7118.5ft a) Angle of elevation = 210 b) Angle of elevation = Record the given angle of elevation on each diagram. What is 3.5m 13.6m 13m a) c) b) d) 18mH measure of each angle of elevation, to the nearest degree? each unknown side length, to one decimal place? 1. Label the angle of elevation, x, on each diagram. What is the C) 8.4 Practice

7 Hint Chapter 9 So/uing Right Trang!e Problems Draw a diagram. What is the height of the statue? At 6.3 rn from the base, the angle of elevation to the top is 6. The statue of Smokey the Bear is a landmark in Reveistoke. angle of elevation be for spring and fall? the brace at a height of 0.92 m. Draw a diagram. What will the 5. In spring and fall, the panels from Question 4 will tilt to meet supporting brace be? are 1.21 m tall. The roof is flat. How tall should the 1.21 brace elevation for panels En Saskatoon is The panels supporting the angle to match the season. In winter, the angle of 4. Bailey installs solar panels in Saskatoon. She adjusts the middle finger. elbow to the tip of 440 cubits distance from the based on the length. It was ancient unit of A cubit is an world. What is the slant height, h, of the face, in cubits? The pyramid is one of the seven wonders of the ancient 3. This diagram shows one face of the Great Pyramid of Giza.

8 a different way? sove this prob:ern looking down at Hint How could you of 180. triangle have a sum line of sight when horizontal and the an object the angle depression angle of. a ruler The angles in a (You will need between the J1 -c /Jcrf; ce the crew to the hiker? Is your answer reasonable? Expiain. The crew reeds to go ft to reach the hker. A search-and-rescue helicopter spots xft= ft How far from the hiker is the ground crew? x= x ground crew from the hiker? = 3 What equation can you use to determine the distance from r-i angles. Label these on the diagram. Use the given angle to calculate the measures of the other How far, to the nearest foot, is the crew ground hiker directly above a ground rescue crew. of 310. The helicopter is flying 1100 ft a lost hiker at an angle of depression -fl ii) What is the angle of depression? ::: C diagram. Label the distance from the crew to the hiker x ft. ) Label the helicopter height and the angle of depression on the L in the diagram. i) Name the angle of depression 1 Angles of Depression.4,. object deprsi > angle of NE -

9 high is the cliff? Luc is looking down at an angle of depression of 72. About how distance between them is 390 m. Marcel, M, at ground level on the far shore. The horizontal D. The angle of depression is about Chapter 9 S(itng Rignt riancjle Problems e3 NE. x = i( x = C. What is an equation for determining the value of x? of depression x. Label the known distances. Label the angle B. Draw a diagram of the plane s descent. 5.7 km in metres: 5.7 km = m A. For an equation, lengths are written with the same unit. Express Solution 5.7 km. What is the angle of depression for her path of descent? cover. the ground. At this point, her horizontal distance to the airport is A pilot begins her final descent to an airport from 300 m above Use the charts inside the back Example 2 Hint The cliff is about high. always happen? you think this will Explain. Marcel s angle depression and C. The equation for the height, h, of the cliff is of elevation? Do 72 = Luc s angle of notice about What did you the diagram. B. What is the angle of elevation for Marcel? Add this to A. Label the given lengths and angles on the diagram. Label the height of the cliff h m. M Solution Luc, L, is on a cliffside trail above a river. He sees his brother Example I

10 252 Ap;,ericeship and Iiorkplace 11 9,0 m angle of depression should Fiona aim the light? 3. Fiona works for a theatre lighting crew. A light has to be 8.2 in. b) 2.. What are the unknown side lengths, to one decimal place? is about \E1 0-1( 41 ft a) b) measure on the diagram. 1. What is the measure of each angle of depression? Label this Practice aimed at an area shown by point P in the diagram. At what w z a) - C D 18.4 m 13.9 m. is about The angle of depression The angle of depression xo =

11 To determine the height of the ride, William rides to the top. \EL Capr 9 SoIvir?g Right rge Prab1e is 253 problems about angles of depression? 6. How can knowing about angles of elevation help you solve tower base is the bear? Give your answer to the nearest foot. Draw a diagram to represent the situation. How far from the spots a bear at an angle of depression of A fire tower in Alberta is 120 ft high. An observer in the tower 13.0 m 1.5m( Pam [3 b) Pam s eyes are 1.5 m above the ground. Is the roller / coaster too tall for the stadium? a) At what angle of depression would William look down from / the top to see Pam? / Wililam. / Pam stands 13 m from the bottom and looks up 73 to see / stadium 56 m high. the next city, the roller coaster is to be set up inside a domed 4. Wililam and Pam are ride operators for a travelling midway. In William

12 24 App Js-rj rc f/d,c2ace 1 B. Sum for each side: 10 D. Sum for each side: 12 get the sum shown for each side. Use each number only once per I A. Sum for each side: 9 C. Sum for each side: 11 can move around. Arrange the numbers 1, 2, 3, 4, 5, and 6 around each triangle to triangle. Explain your strategy. paper numbers you Try making some Hint Solving Triangle Puzzles

13 Mid-Chapter 4 4, Chapter 9 S&vinq Right Than gie rhlems 255 away from the Calgary Tower. How tall is the building? depression to a nearby building as 40. The building is 61 m m above the ground. She estimates the angle of 4. An observer is on the Calgary Tower Observation Deck, stringer r elevation, x, of the stringer, to the nearest degree? 1 a riser of 160 mm and tread of 290 mm. What is the angle of 3. Yanek builds custom decks. The stairs for a customer have riser /1 How high above the ground is the kite? 2. Aiko is flying a kite. She is using a string that is 95 ft long. a) sin 72 b) cos 35 = 1. What is the value of x, to one decimal place?

14 x W Hhit How tall is the inuksuk, to the nearest tenth of a metre? fl x =, What. i) Name all the right triangles in the diagram. M pictures of the tourists by the giant inuksuk. Nuvuk works as a tour guide in Rankin Inlet. He often takes Q R S p You will need 9A Solving TwomTriangle Prob ems _- 256 i nz s iri //2cace xpan. reasonab e? ansvier s your fnal inuksuk. the height of the height. IG shows MG shows Milan s (L_J aruler The inuksuk s tall, to the nearest tenth of a metre. = x, or x cm O How tall s the inuksuk The dstance IS - cm y=.,or centre of the base of the inuksuk? is the distance from P to the Label the inuksuk height x cm. known angles and side lengths. Label the diagram to show the was 26. Nuvuk took a picture of Milan. Milan is 187 cm tall. From point P, the angle of elevation to the top of Milan s head The angle of elevation to the top of the inuksuk was 500. SoPQ= cm. ii) PR = 25cm and QR = 16cm. G P y cm

15 7.8 km to Jarvis Bay Provincial Park, J. They leave Half Moon Bay, H. They paddle canoe trip across Sylvan Lake. j CpEr9 Slilr Right, &ngle Problerrs 257 They shoud turn at an angle of about 7.6 LB 7.6 D. At what angle should the canoeists turn? B C. Calculate the length of h. triangles are AHXJ and AHXB? Show this on the diagram. B. Draw a height, h, from H to point X on BJ. What kind of diagram. At each step, triangle? Show how you know. Hint the lengths and angles. Is z \HJB a right update the A. Draw the triangle that maps their route. Label km Solution H where they started? At what angle should they turn in order to return to Moon Sylvan Lake from Half Moon Bay. beach. This beach, B, is 7.6 km across the lake Half Then they turn 69. They paddle 5.1 km to a Park 1 m A camp instructor leads a group of students on a N a Example

16 PhiIli...j3O0 H To calculate his angle of The final distance from Ann to Julian is 9.2 m. 28 Apprenticeship and Jirh place 11 4EL peripheral vision, Cal asks C person sees about look straight ahead. An average know. B A the information you the diagram with determines how far you can Start by labelling 2. Your angle of peripheral vision Hint... S..5 S 44 72m From H to T: From B to H: a) What is the height of each part of the pillar? b) How many metres tall is the pillar from bottom to top? / Lion s Gate Bridge in Vancouver. Julian and Ann to walk in opposite directions from point X. Julian and Ann have each walked the same distance. What is Cal s angle of peripheral vision, LACJ? Julian and Ann stop when Cal can no longer see them. He watches from point C, 3.0 m away from X. - T 1. PhiHip is on the shore looking at a pillar, BT, on the Practice see from left to right when you X.Om 7 () 9.2 m

17 a) What is the distance from the ground, G, to each point? The angle of elevation to the top of the nickel is 766. the angle of elevation to the bottom of the nickel is From point P, 10 ft away from the coin s base on the ground, 3. The Big Nickel in Sudbury is the world s largest coin. Na Chapter.9 ScIving Right TriavgIe rcb1erns 259 Draw a diagram with your solution. the bridge with a 140 wide-angle lens? the bridge. 600 m away horizontally from one end of The helicopter will be 370 m above and The bridge is 1517 m long. Can Anya capture the entire distance across -rid a helicopter. to photograph the Lion s Gate Bridge from 4. Anya is an aerial photographer. She plans b) What is the height of just the nickel, to the nearest foot? loft nickel, B: nickel, T: Height to bottom of Height to top of

18 x x =AB =AB AB CB 9 and A - aruler You wil need Draw a line to form two triangles that you A ( Solving 3-0 Triangle Problems 8 in =CB =CB tan 0 z D zv en:cd3d //-p The cliff is high, to the nearest metre. trar p e? a secor d rgh from Car to make e.p to waik away Thr.k t woud Vhy dd Devon How high s the chif, to the nearest metre? the cliff at B? How far is it from where Carl is standing at C to the base of ) Name the right triangles in the diagram. How high is the cliff, to the nearest metre? at a right angle to BC. LBDC is 78. From Carl, C, the angle of elevation to Devon, D, walks 105 m away from Carl, the base, B, to the top, A. the cliff top, L4CB, is 510. second peak of Stawamus Chief, from They need to determine the height of the Carl and Devon are rappelling instructors. AY, of the box. Name the triangles. could use to determine the length of diagonal, 9in. y

19 How far apart are the two boats? sees a Coast Guard boat at C. level. Ranaan sees a boat in distress at D. He The top of the Hghthouse is 160 ft above sea 160 ft Chapter 9 SIirq Rih. Triançle Pmblerns 261 B x 0 on the diagram. What is the slant height, HX? Mark the measurements a square base with m sides and is m high. 1. A pyramid at the Muttart Conservatory in Edmonton has H Practice The boats are about ft apart. = z2= + C. Calculate the distance between the two boats. z2=x 2+y 2 tan x X tan 38 = 160 yxtan24 = E1 tan Coast Guard boat: Boat in distress: wfth each part. B. Calculate the horizontal distance from point B to each boat. Update the diagram Hint A. Name all the right triangles in the diagram. Solution Ranaan works in the Langara Point Lighthouse. Example

20 262 Apprenticeship and Workplace 1 1 NE.. A To get LHAP: cos LHAP =, or B X 2 2 H b) Did Judy use the correct trigonometric ratio? Explain. a c) Was Judy correct to determine LHAP? Explain. a) Was the length Judy calculated for AC reasonable? Explain. each face of the pyramid. Is her solution 13.0 The angle o4! elevation is abo.i Judy calculated the angle of elevation of = , or m A 18.0 m high. base with sides 19.5 m long. They are 19.5 m B D pyramids. Two pyramids have a square 2. The Muttart Conservatory has four H the diagram. Mark the measurements on to one decimal place? for the pyramid in Question 2, 3. What is the angle of elevation Len 9l+, o4! AP: = AC, or AC = m.4p is hal4! 4! AC = AC, C 2 = AC reasonable?

21 45cm Diagonal length AB: Inside diagonal AC: B Assume that the tubing does not bend. A 30cm tubing he can pack in the box shown? Ci.spter 9 Sci itrg Rqht friangle Problems 263 the tower more than, less than, or equal to 10? Explain. b) Is the angle of elevation from Edona s house to the top of a) How tall is the tower, to the nearest metre? the top is 10. is due west of Amy s house. house, travelling 400 east of north. The tower Edona s house. Edona walks 1.5 km to Amy s The angle of elevation from Amy s house to A (Amy) N T tower 5. A communications tower lies due north of H b) What is the measure of ZCAB in Part a)? What is the longest piece of thin copper 4. a) Ryan runs an onilne parts supply business.

22 e4 r:sfjc1 C to the nearest foot? a) What is the maximum safe angle of elevation for x ft. How high is the tail, y distance from the dinosaur, was the same horizontal on the ground. Each point head and fts tail from points the angles of elevation to its is 86 ft tah. Anita measured dinosaur, in Drumheller, 2. The world s largest b) Brynn s ladder is 24 ft long. How high can a 24 ft ladder least 1 ft away from the wall for every 4 ft of ladder length. regulations state that the base of a ladder must be at a ladder, without rounding? reach, without rounding? calculation. x x angle cannot be the result of the 56 for a 4 ft ladder. the same as it is Hint for any ladder is angle of elevation greater or ess than maximum safe The maximum safe Part a), the For Question 1 Hint 1. Brynn installs soffit, fascia, and eavestrough. Safety Chapter ic

23 The wires meet the pole at 20 ft and 40 ft above the ground. chapter 9 Solvnq Sight Tri&ngle Problems 265 similar? How are they different? 5. How are the angles of elevation and angles of depression nearest metre? is the river, w, to the The diagram shows on shore BD. How wide ACDE measured out Winnipeg, surveyors 48 m 20 C width across the river. needed to determine the 82 m B Pedestrian Bridge in Esplanade Riel 4; To construct the depression of each wire, to the nearest degree? Label the lengths and angles you know. What is the angle of Draw a diagram to show the angle of depression of each wire. The wires are anchored at a point 24 ft from the pole base. on rocky ground, she uses guy wires to secure the pole. 3. Alexis installs wind turbines on the Prairies. For installations

24 77 nearest metre? Brooke stands between two poles. She is 25 ft from the The angle of elevation from Brooke s feet to the top of long is the zip line, to the Pole 1 is m below the starting point on the far side. How must land at a distance of line along the line BD. It 2. Phuong is instaliirg a zip C-. poles, to the nearest foot? a) Draw a diagram for the problem. How tall is Pole 1? Brooke s feet is 19. A linesman is on top of Pole 2. The angle of depression to b) The two poles are the same height. How far apart are the I base of Pole Brooke replaces old wooden utihty poles with steel ones. Chapter 11