Pre-Publication110˚ · 30 GPS satellites. Also in 2010, Russia had almost completed its own system of 24 satellites, and both the European Community and China were beginning to deploy

NEL158

51 yards

160 yards

110˚

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NEL 159

LEARNING GOALS

You will be able to develop your spatial sense by

• Using the sine law to determine side lengths and angle measures in obtuse triangles

• Using the cosine law to determine side lengths and angle measures in obtuse triangles

• Solving problems that can be modelled using obtuse triangles

This scenic hole at Furry Creek golf course near Vancouver has a dogleg left. On a dogleg hole, golfers have a choice between playing it safe and making the green in two shots or taking a chance and trying for the green in one shot. Jay can hit a ball between 170 and 190 yd from the tee with a 3-iron. Is it possible for Jay to make the green at this hole in one shot with a 3-iron? Explain.

4Chapter

?

Oblique Triangle

Trigonometry

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NEL160 Chapter 4 Oblique Triangle Trigonometry

4 Getting Started

GPS Trigonometry

Global Positioning System (GPS) satellites were first used by the U.S. Navy in the 1960s. At first, the U.S. Navy had only five satellites and was able to receive precise navigational locations only once every hour. As well, the signal was intentionally altered for civilian use, so that only the military would have access to the full precision of the system. In 2000, civilians also gained access, bringing the available precision of the GPS system from 1000 ft to 65 ft.

By 2010, the GPS system in the United States had expanded to include 30 GPS satellites. Also in 2010, Russia had almost completed its own system of 24 satellites, and both the European Community and China were beginning to deploy GPS systems. Europe and China say their systems will be precise to 10 m.

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NEL 161Getting Started

Two of the 10 GPS tracking stations in Canada are located in Victoria, British Columbia, and Prince Albert, Saskatchewan, 1340 km apart. Suppose that these stations locate a GPS satellite at the same time, when the satellite is vertically above the line segment connecting them. The angles of elevation from the tracking stations to the satellite measure 87.7° and 88.5°.

What is the altitude of the GPS satellite?

A. Draw a triangle that models the two tracking stations and the satellite. What type of triangle did you draw?

B. Solve your triangle.

C. Create a plan that will allow you to calculate the altitude of the satellite.

D. Carry out your plan to determine the altitude of the satellite.

WHAT DO You Think?Decide whether you agree or disagree with each statement. Explain your decision.

1. Each value of a primary trigonometric ratio corresponds to one unique angle.

2. The sine law and cosine law relationships apply only to angles and sides in acute or right triangles.

3. When modelling a problem that can be solved using trigonometry, the information provided will lead to only one possible triangle.

?

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4.1 Exploring the Primary Trigonometric Ratios of Obtuse Angles

YOU WILL NEED• calculator

Determine the relationships between the primary trigonometric ratios of acute and obtuse angles.

EXPLORE the MathUntil now, you have used the primary trigonometric ratios only with acute angles. For example, you have used these ratios to determine the side lengths and angle measures in right triangles, and you have used the sine and cosine laws to determine the side lengths and angle measures in acute oblique triangles.

Joe investigated the values of the primary trigonometric ratios for obtuse angles. Using a calculator, he determined that the value of sin 100° is 0.9848… .

He knew that he could not create a right triangle with a 100° angle. However, he knew that he could create a triangle using the supplement of 100°, which is 80°. Out of curiosity, he evaluated sin 80° and determined that it has the same value, 0.9848… .

Joe decided to broaden his investigation. He created a table like the one below.

B

A

C

ab

c

sine lawa

sin A5

b sin B

5c

sin C

cosine law

a25b21c222bccosA

100°80°

oblique triangle

Atrianglethatdoesnotcontaina90°angle.

GOAL

What relationships do you observe when comparing the trigonometric ratios for obtuse angles with the trigonometric ratios for the related supplementary acute angles?

?

u sin u cos u tan u (180° 2 u) sin (180° 2 u) cos (180° 2 u) tan (180° 2 u)

100° 0.9848 80° 0.9848

110°120°130°

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NEL 1634.1 Exploring the Primary Trigonometric Ratios of Obtuse Angles

Reflecting

A. Compare your observations with a classmate’s observations. How are they different? How are they alike?

B. Describe any patterns you observed as the measure of the obtuse angle increased.

FURTHER Your Understanding 1. Which of the following equations are valid? Give reasons for your

choices.a) sin 25° 5 sin 65° d) sin 122° 5 sin 58°b) cos 70° 5 2cos 110° e) cos 135° 5 cos 45°c) tan 46° 5 tan 134° f ) tan 175° 5 2tan 5°

2. Calculate each ratio to four decimal places. Predict another angle that will have an equal or opposite trigonometric ratio. Check your prediction.a) sin 15° c) tan 35°b) cos 62° d) sin 170°

3. Determine two angles between 0° and 180° that have each sine ratio.a) 0.64 c) 0.95

b) 13

d)7

23 4. a) Identify pairs of angles with equal sine ratios in the five triangles to

the right.b) What do you know about the cosine and tangent ratios for these

pairs of angles?

In Summary

Key Idea

• There are relationships between the value of a primary trigonometric ratio for an acute angle and the value of the same primary trigonometric ratio for the supplement of the acute angle.

Need to Know

• For any angle u, sin u 5 sin (180° 2 u) cos u 5 2cos (180° 2 u) tan u 5 2tan (180° 2 u)

80°

55°

A

DE

N

M O

LK

J

F

G

I

H

B C

110°

20°

40°

20°

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4.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles

YOU WILL NEED• calculator• ruler Explain steps in the proof of the sine and cosine laws for obtuse

triangles, and apply these laws to situations that involve obtuse triangles.

INVESTIGATE the MathIn Lesson 3.2, you analyzed Ben’s proof of the sine law for acute triangles. Ben wanted to adjust his proof to show that the sine law also applies to obtuse triangles. Consider Ben’s new proof:

Step 1 I drew obtuse triangle ABC with height AD.

Step 2 I wrote equations for sin (180° 2 / ABC ) and sin C using the two right triangles.

In ^ ABD,

sin (180° 2 / ABC) 5 opposite

hypotenuse

sin (180° 2 / ABC) 5 ADc

c sin (180° 2 / ABC) 5 AD c sin / ABC 5 AD

In ^ ACD,

sin C 5 opposite

hypotenuse

sin C 5 ADb

b sin C 5 AD

Step 3 Both expressions for AD equal each other (transitive property), so: c sin / ABC 5 b sin C

1 c sin /ABC 2

sin C 5 b

c

sin C5

bsin /ABC

Step 4 I drew a new height, h, from B to base b in the triangle. In ^ ABE, In ^ CBE,

sin A 5hc sin C 5

ha

c sin A 5 h a sin C 5 h

Step 5 Both expressions for h equal each other, so: c sin A 5 a sin C

c

sin C5

a sin A

D B

c

a

b

C

A

EXPLORE…

• Anisoscelesobtusetrianglehasoneanglethatmeasures120°andonesidelengththatis5m.Whatcouldtheothersidelengthsbe?

GOAL

D B

c

a C

A

E

h

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NEL 1654.2 Proving and Applying the Sine and Cosine Laws for Obtuse Triangles

I have already shown that c

sin C5

b sin /ABC

, so

c

sin C5

b sin /ABC

5a

sin A

How can you explain what Ben did to prove the sine law for obtuse triangles?

A. Why did Ben choose to write expressions for the sin (180° 2 / ABC) and sin C ?

B. In step 3, Ben mentions the transitive property. What is this property, and how did he use it in this step?

C. In step 4, Ben drew a new height in ^ ABC. Why was this necessary?

D. Why was Ben able to equate all three side angle ratios in step 5?

Reflecting

E. Compare the proof above to Ben’s original proof in Lesson 3.2, pages 118 to 119. How is the proof of the sine law for obtuse triangles the same as that for acute triangles? How is it different?

F. If Ben started his proof by writing expressions for sin (180° 2 / CBA) and sin A, where would he have drawn the height in step 1?

?

APPLY the Mathexample 1 Use reasoning and the sine law to determine

the measure of an obtuse angle

In an obtuse triangle, /B measures 23.0° and its opposite side, b, has a length of 40.0 cm. Side a is the longest side of the triangle, with a length of 65.0 cm. Determine the measure of / A to the nearest tenth of a degree.

Bijan’s Solution

b 40.0 cma 65.0 cm

23°

C

A B

I drew an obtuse triangle to represent ^ ABC.

I knew that the longest side is always opposite the largest angle, so the 65.0 cm side must be opposite the obtuse angle, / A.

Since ^ ABC is not a right triangle, I knew that I could not use the primary trigonometric ratios to determine the measure of /A.

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sin Aa 5

sin Bb

sin A65.0

5sin 23°

40.0

65.0a sin A65.0

b 5 a sin 23°

40.0b65.0

sin A 5 0.6349…

/ A 5 sin−1(0.6349…) / A 5 39.4153…°

/ A 5 180° 2 39.4153…°/ A 5 140.5846…°

/ A measures 140.6°.

Your Turn

Determine the length of side AB in ̂ ABC above, to the nearest tenth of a centimetre.

Inoticedthatthediagramhastwoside-anglepairswithonlyoneunknown,/A.Idecidedtousethesinelaw.

Myreasoningsuggeststhat/ Amustbetheobtuseangle.IusedtherelationshipsinA5sin(180°2A).

Themeasureofanangleistheunknown,soIusedtheformofthesinelawthathastheanglesinthenumerator.

IisolatedsinA.

Themeasureoftheangleseemsappropriate,accordingtomydiagram.

example 2 Solving a problem using the sine law

Colleen and Juan observed a tethered balloon advertising the opening of a new fitness centre. They were 250 m apart, joined by a line that passed directly below the balloon, and were on the same side of the balloon. Juan observed the balloon at an angle of elevation of 7o while Colleen observed the balloon at an angle of elevation of 82o. Determine the height of the balloon to the nearest metre.

Colleen’s Solution

DJ

B

C 82°

7°

250 m

Idrewadiagramtorepresent the situation.Theheightoftheballoonisrepresented by BD. I needtodetermine thelengthofBC in order to determinethelengthofBD.Icanusethe sine law in ∆BJC.

Iusedtheinversesinetodeterminethemeasureof/ A.

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example 3 Use reasoning to demonstrate the cosine law for obtuse triangles

Show that the cosine law holds for obtuse triangles, using ^ ABC.

B

A C

a

b

c

Hyun Yoon’s Solution

D

h

ACB

180° ACB

a

B

A Cb x

c

IextendedthebaseofthetriangletoD.Thiscreatedtwooverlappingrighttriangles,^CBDand^ ABD,withheightBD.ItalsocreatedtwoanglesatC,/ ACBand/DCB,suchthat/DCB5180°2/ ACB.

/BCJ 5 180° 2 82°/BCJ 5 98°

/JBC 5 180° 2 98° 2 7°/JBC 5 75°

BC sin /BJC

5JC

sin /JBC

BC sin 17° 2 5

250 sin 175° 2

BC 5 sin 17° 2 a 250 sin 175° 2 b

BC 5 31.542....

sin 1/BCD 2 5BDBC

sin (82°) 5 BD

31.542...(31.542...) (sin (82°)) 5 BD

31.235... m 5 BDThe advertising balloon is 31 m above the ground.

Your Turn

Determine the distance between Juan and the balloon.

Ideterminedthesupplementof82°todeterminethemeasureofasecondanglein^ BJC.Thisisanobtusetriangle.

Ideterminedthemeasureofthethirdanglein∆BJC.Thisgave meaknownside,JC,andaknownangleoppositethis side,/JBC,inthistriangle.

IusedthesinelawtowriteanequationthatinvolvedBCandtheknownside-anglepair.

IsubstitutedtheknowninformationintotheequationandsolvedforBC.

IsubstitutedtheknowninformationintotheequationandsolvedforBD.

IwroteanequationthatinvolvedBD,BC,andtheknownanglein∆BCD.

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In ^ ABD In ^CBDh2 5 c2 2 (b 1 x)2 h2 5 a2 2 x 2

c 2 2 (b 1 x)2 5 a2 2 x 2 c 2 5 (b 1 x)2 1 a2 2 x 2 c 2 5 b2 1 2bx 1 x 2 1 a2 2 x 2 c 2 5 a2 1 b2 1 2bx

cos (180° 2 / ACB) 5 xa

a cos (180° 2 / ACB) 5 x

c 2 5 a2 1 b2 1 2b[a cos (180° 2 / ACB)]

c 2 5 a2 1 b2 2 2ab cos / ACBI have demonstrated the cosine law.

Your Turn

Review the proof of the cosine law for acute triangles in Lesson 3.3, pages 130 to 131. Explain how Hyun Yoon modified this proof to deal with an obtuse triangle.

example 4 Using reasoning and the cosine law to determine the measure of an obtuse angle

The roof of a house consists of two slanted sections, as shown. A roofing cap is being made to fit the crown of the roof, where the two slanted sections meet. Determine the measure of the angle needed for the roofing cap, to the nearest tenth of a degree.

Maddy’s Solution: Substituting into the cosine law and then rearranging

c 33.5 ft

b 20.3 fta 17.0 ft

c2 5 a2 1 b2 2 2ab cos u

roofing cap

17.0 ft

33.5 ft

20.3 ft

Isketchedatriangletorepresenttheproblemsituation.

Thelargestangleisu,becauseitisoppositethelongestside.

Threesidelengthsaregiven,soIknewthatIcouldusethecosinelaw.

IusedthePythagoreantheoremtowritetwoexpressionsforh2,usingthetworighttriangles.

Theexpressionsthatequalh2equaleachother(transitiveproperty).

Isolvedforc2.

Theacuteanglein^CBDhasameasureof180°2/ ACB.

Iusedthecosineratiotowriteanexpressionforx.

Isubstitutedmyexpressionforxintomyequation.

Towriteanequationthatcontainedonlymeasuresfoundintheoriginaltriangle,Iusedthefollowingfact:cos(180°2/ ACB)52cos/ ACB

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Isubstitutedthevaluesofa,b,andcintotherearrangedformula.

SinceIwantedtosolveforu,Irearrangedtheformulatoisolatecosu.

(33.5)2 5 (17.0)2 1 (20.3)2 2 2(17.0)(20.3) cos u(33.5)2 2 (17.0)2 2 (20.3)2 5 22(17.0)(20.3) cos u

1122.25 2 289 2 412.09 5 −690.2 cos u

421.162 690.2

5 cos u

cos−1a2 421.16690.2

b 5 u

127.6039…° 5 u

An angle of 127.6° is needed for the roofing cap.

Georgia’s Solution: Rearranging the cosine law before substituting

c 33.5 ft

b 20.3 fta 17.0 ft

c2 5 a2 1 b2 2 2ab cos u

c2 1 2ab cos u 5 a2 1 b2 2 2ab cos u 1 2ab cos u

c2 2 c2 1 2ab cos u 5 a2 1 b2 2 c2

2ab cos u2ab

5a2 1 b2 2 c2

2ab

cos u 5a2 1 b2 2 c2

2ab

cos u 5117.0 22 1 120.3 22 2 133.5 22

2 117.0 2 120.3 2cos u 5 20.6101… u 5 cos21(20.6101…) u 5 127.6039…°

The angle for the roofing cap should measure 127.6°.

Your Turn

Determine the angle of elevation for each roof section, to the nearest tenth of a degree.

Isubstitutedtheknownvaluesintotheformulaforthecosinelawandisolatedu.

Myanswerisreasonable,giventhediagram.

Isketchedatriangletorepresenttheproblemsituation.

Iknewthelengthsofallthreesides,soIusedthecosinelaw.

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In Summary

Key Idea

• Thesinelawandcosinelawcanbeusedtodetermineunknownsidelengthsandanglemeasuresinobtusetriangles.

Need to Know

•Thesinelawandcosinelawareusedwithobtusetrianglesinthesamewaythattheyareusedwithacutetriangles.

Use the sine law when you know …

Use the cosine law when you know …

-thelengthsoftwosidesandthemeasureoftheanglethatisoppositeaknownside

-thelengthsoftwosidesandthemeasureofthecontainedangle

-themeasuresoftwoanglesandthelengthofanyside

or

-thelengthsofallthreesides

•Becarefulwhenusingthesinelawtodeterminethemeasureofanangle.Theinversesineofaratioalwaysgivesanacuteangle,butthesupplementaryanglehasthesameratio.Youmustdecidewhethertheacuteangle,u,ortheobtuseangle,180°2u,isthecorrectangleforyourtriangle.

•Becausethecosineratiosforanangleanditssupplementarenotequal(theyareopposites),themeasuresoftheanglesdeterminedusingthecosinelawarealwayscorrect.

CHECK Your Understanding 1. There are errors in each application of the sine law or cosine law.

Identify the errors.a) b)

5sin 100°

5 x

sin 32°122 5 x2 1 122 2 2(12)(x) cos 115°

100° 5 m

x32°

12 cm115°

12 cm

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2. Which law could be used to determine the unknown angle measure or side length in each triangle? For your answer, choose one of the following: sine law, cosine law, both, neither. Explain your choice.a) d)

b) e)

c)

PRACTISING3. Determine the unknown side length in each triangle, to the nearest tenth

of a centimetre.

15 m

12 m 95°

x

35°

28°33 in.

x

5 cm

7 cm

3 cmx

x150°

18°12°

110° 22 m25 m

x

4. Determine the unknown angle measure in each triangle, to the nearest degree.

a)

4.0 cm101.0°

x

28.0°

b) c)

130.0°

1.4 cm

2.0 cm

x

32.0°30.0 cm

24.0 cm x

118°

68 m

44 m

x

4 cm

5 cm2 cm

x

150°

180 cm106 cm x

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5. Determine each unknown angle measure to the nearest degree and each unknown side length to the nearest tenth of a centimetre.

a) c) A C

B

5 cm

10 cm8 cm

b) d)

6. A triangle has side lengths of 4.0 cm, 6.4 cm, and 9.8 cm.a) Sketch the triangle, and estimate the measure of the largest angle.b) Calculate the measure of the largest angle to the nearest tenth of

a degree.c) How close was your estimate to the angle measure you calculated?

How could you improve similar estimates in the future?

7. Wei-Ting made a mistake when using the cosine law to determine the unknown angle measure below. Identify the cause of the error message on her calculator. Then determine u to the nearest tenth of a degree.

10

20

12 202 5 102 1 122 2 2(10)(12) cos u 400 5 100 1 144 2 240 cos u 400 5 244 2 240 cos u 400 5 4cos u 100 5 cos ucos−1(100) 5 u <error!> 5 u

8. In ^QRS, q 5 10.2 m, r 5 20.5 m, and s 5 12.8 m. Solve ^QRS by determining the measure of each angle to the nearest tenth of a degree.

9. While golfing, Sahar hits a tee shot from T toward a hole at H. Sahar hits the ball at an angle of 23° to the hole and it lands at B. The scorecard says that H is 295 yd from T. Sahar walks 175 yd to her ball. How far, to the nearest yard, is her ball from the hole?

105°7.5 cm

11.2 cmN

M

L

TS

R

28°120°

25.6 cmY Z

X

18.7 cm35°21°

20.5 m12.8 m

10.2 m

Q

SR

23°T

B

H

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10. The posts of a hockey goal are 6 ft apart. A player attempts to score by shooting the puck along the ice from a point that is 21 ft from one post and 26 ft from the other post. Within what angle, u, must the shot be made? Express your answer to the nearest tenth of a degree.

11. In ^DEF, /E 5 136°, e 5 124.0 m, and d 5 68.4 m. Solve the triangle. Round each angle measure or side length to the nearest tenth.

12. A 15.0 m telephone pole is beginning to lean as the soil erodes. A cable is attached 5.0 m from the top of the pole to prevent the pole from leaning any farther. The cable is secured 10.2 m from the base of the pole. Determine the length of the cable that is needed if the pole is already leaning 7° from the vertical.

13. A building is observed from two points, P and Q, that are 105.0 m apart. The angles of elevation at P and Q measure 40° and 32°, as shown. Determine the height, h, of the building to the nearest tenth of a metre.

14. A surveyor in an airplane observes that the angles of depression to points A and B, on opposite shores of a lake, measure 32° and 45°, as shown. Determine the width of the lake, AB, to the nearest metre.

Closing

15. In ^PQR, /Q is obtuse, /R 5 12°, q 5 15.0 m, and r 5 10.0 m. Explain to a classmate the steps required to determine the measure of /Q.

Extending

16. Two roads intersect at an angle of 15°. Darryl is standing on one of the roads, 270 m from the intersection.a) Create a problem that must be solved using the sine law.

Include a sketch and a solution.b) Create a problem that must be solved using the cosine law.

Include a sketch and a solution.

17. The interior angles of a triangle measure 120°, 40°, and 20°. The longest side of the triangle is 10 cm longer than the shortest side. Determine the perimeter of the triangle, to the nearest centimetre.

68.4 m 136°

124.0 m DF

E

15.0 m

5.0 m

cable

10.2 m

105.0 mQ

P

h 32°

40°

9750 m

A B

32°45°

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4FREQUENTLY ASKED QuestionsQ: How are the primary trigonometric ratios for an obtuse

angle related to the primary trigonometric ratios for its supplementary acute angle?

A: The primary trigonometric ratios for supplementary angles (one being acute and the other being obtuse) are either equal or opposite. The sine ratios for supplementary angles are equal. The cosine and tangent ratios for supplementary angles are opposites.

30°150°

For example, 150° and 30° are supplementary angles.

sin 150° 5 0.5 cos 150° 5 20.8660... tan 150° 5 20.5773...

sin 30° 5 0.5 cos 30° 5 0.8660... tan 30° 5 0.5773...

Q: Why do you sometimes need a diagram when you are using the sine law to determine the measure of an angle, whereas you do not need a diagram when you are using the cosine law?

A: When you use the sine law or the inverse sine to determine the measure of an angle, there are always two possibilities for the measure:

sin u 5 sin (180° 2 u)

The angles that give equal sine ratios are supplementary, so you may need to check a diagram or interpret the problem carefully to determine which angle measure is appropriate.

When you use the cosine law or the inverse cosine to determine the measure of an angle, there is only one possible measure:

cos u 5 2cos (180° 2 u)

Study Aid•SeeLesson4.1.•TryMid-ChapterReviewQuestions1to3.

Mid-Chapter Review

Study Aid•SeeLesson4.2,Examples1,2,and4.

•TryMid-ChapterReviewQuestions4to9.

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NEL 175Mid-Chapter Review

Lesson 4.1

1. Determine each ratio to four decimal places. Then identify another angle that has an equal or opposite ratio. Verify your answers.a) sin 75° d) sin 172°b) cos 100° e) cos 38.5°c) tan 32° f ) tan 122.3°

2. Draw obtuse triangle ABC and acute triangle DEF to satisfy each of the following conditions:a) No internal angles have the same measure.b) Two angles, one from each triangle, have

equal sine ratios.

3. Determine all the angles that satisfy each ratio below (0 , u , 180°). Round the angles to the nearest degree.

a) sin u 5 0.362 d) sin u 5 12

b) cos u 5 20.75 e) cos u 5 0.214

c) tan u 5 52

f ) tan u 5 1

Lesson 4.2

4. Calculate the indicated angle measure or side length in each triangle, to the nearest tenth.

11.0 m

15.0 m

2.5 km

y

155.0°

10.7 m

5. In ^DEF, /E 5 132°, e 5 20 cm, and d 5 15 cm. Determine the measure of /F to the nearest degree.

6. a) Determine the measure of the indicated angle in each triangle, to the nearest tenth of a degree.

i)

46°

3.0

2.7

ii)

2.5

7.1

5.0

b) Which angle measure in part a) could you not have determined without a diagram? Explain.

7. Draw an obtuse triangle, and measure two sides (with a ruler) and one angle (with a protractor).a) If possible, use the sine law to determine the

measure of another angle. If you cannot use the sine law, explain why.

b) Determine a different angle measure another way. Use a protractor to verify the measure.

8. In ^ ABC, / A is obtuse, /C 5 15°, c 5 3.0 cm, and a 5 4.7 cm. Determine the measure of / A to the nearest degree.

9. A boat travels 60 km due east. It then adjusts its course by 25° northward and travels another 90 km in this new direction. How far is the boat from its initial position, to the nearest kilometre?

PRACTISING

a)

b)

2.0 cm

x121.0°

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4.3 The Ambiguous Case of the Sine Law

Analyze the ambiguous case of the sine law, and solve problems that involve the ambiguous case.

INVESTIGATE the MathNaomi works for a company that makes supporting braces for solar panels. She is drawing a scale diagram to show solar panels that are going to be installed on the flat roof of a downtown high-rise. Each panel is 5.5 m long and must be tilted at 40° to the horizontal in order to maximize the strength of the Sun’s rays. Naomi needs to choose the length of the supporting brace for each panel. Supporting braces are available in 1 m increments, starting at 2 m and going up to 6 m.Naomi started with a 2 m brace and discovered that she could not complete the triangle.

40°

supportingbrace

solarpanel

5.5 cm 2 cm

How many different scale diagrams are possible, with the supporting braces that are available?

A. Work with a partner. Use a ruler and a protractor to construct a 40° angle connected to a 5.5 cm side, as shown in Naomi’s diagram.

B. Calculate the height of any triangle formed using the 5.5 cm side and 40° angle.

C. The 2 cm side is too short. Try side lengths from 3 cm to 6 cm. The side that is opposite the 40° angle can be at any angle to the 5.5 cm side.

D. What length of supporting brace is necessary in order to have two possible triangles? Explain.

?

GOALYOU WILL NEED• calculator• ruler• protractor

EXPLORE…

• Twosidesinanobtusetriangleare3mand4minlength.Theanglethatisoppositethe3msidemeasures40°.Determinethemeasureoftheanglethatisoppositethe4mside.

Idrewascalediagramofthesituation,using1cmtorepresent1m.Idrewanangleof40°first.ThenImeasured5.5cmalongoneofthearmstorepresentthesolarpanel.

Idon’tthinkatrianglewiththesemeasurementsexists.

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NEL 1774.3 The Ambiguous Case of the Sine Law

Reflecting

E. What range of supporting brace lengths result in two possible triangles?

F. What information was Naomi originally given? Will this type of information always lead to the ambiguous case of the sine law ?

G. When dealing with a SSA situation, how does the height of the triangle help you determine the number of possible triangles?

APPLY the Mathexample 1 Connecting the SSA situation to the number

of possible triangles

Given each SSA situation for ^ ABC, determine how many triangles are possible.a) / A 5 30°, a 5 4 m, and b 5 12 m c) / A 5 30°, a 5 8 m, and b 5 12 mb) / A 5 30°, a 5 6 m, and b 5 12 m d) / A 5 30°, a 5 15 m, and b 5 12 m

ambiguous case of the sine law

Asituationinwhichtwotrianglescanbedrawn,giventheavailableinformation;theambiguouscasemayoccurwhenthegivenmeasurementsarethelengthsoftwosidesandthemeasureofananglethatisnotcontainedbythetwosides(SSA).

Saskia’s Solution

b 12

A

h30°

sin 30° 5 h

1212 sin 30° 5 h 6 m 5 h

a) / A 5 30°, a 5 4 m, and b 5 12 m12 m 4 m

A30°

No triangles are possible.

b) / A 5 30°, a 5 6 m, and b 5 12 m12 m

6 m

A30°

One triangle is possible.

Idrewthebeginningofatrianglewitha30°angleanda12mside.

Iusedthesineratiotocalculatetheheightofthetriangle.

Ican usethisheightasabenchmarktodecideonsidelengthsoppositethe30°anglethatwillresultinzero,one,ortwotriangles.

Sincea,banda,h,Iknewthatnotrianglesarepossible.

Iusedacompasstobecertain.Isetthecompasstipstorepresent4m.Iplacedonetipofthecompassattheopenendofthe12msideandswungthepenciltiptowardtheotherside.Thepencilcouldn’treachthebase,soa4msidecouldnotclosethetriangle.

Sincea,banda5h,thereisonlyonepossibletriangle,arighttriangle.

Acompassarcintersectsthebaseatonlyonepoint.

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c) / A 5 30°, a 5 8 m, and b 5 12 m12 m

8 m

A30°

Two triangles are possible.

d) / A 5 30°, a 5 15 m, and b 5 12 m

A30°

15 m12 m

One triangle is possible.

Your Turn

Determine how many triangles are possible, given / A 5 120°, a 5 15 m, and b 5 12 m.

example 2 Solving a problem using the sine law

Martina and Carl are part of a team that is studying weather patterns. The team is about to launch a weather balloon to collect data. Martina’s rope is 7.8 m long and makes an angle of 36.0° with the ground. Carl’s rope is 5.9 m long. Assuming that Martina and Carl form a triangle in a vertical plane with the weather balloon, what is the distance between Martina and Carl, to the nearest tenth of a metre?

Sandra’s Solution: Using the sine law and then the cosine law

Let h represent the height of the weather balloon. Let u represent the angle for Carl’s rope.Situation 1:

36.0°

7.8 m5.9 m

MartinaCarl

balloon

h

Situation 2:

36.0°

7.8 m5.9 m

MartinaCarl

balloon

Sincea,banda.h,therearetwopossibletriangles.

Acompassarcintersectsthebaseattwopoints.

Sincea.b,onlyonetriangleispossible.

Acompassarcintersectsthebaseatonlyonepoint.

Idrewthetriangle.

InoticedthatthisisaSSAsituation.Ihadtodeterminetheheightofthetriangletodetermineifthisisanambiguouscase.

sin 36.0 5 h

7.87.8(sin 36.0) 5 7.8a h

7.8b

4.5847… 5 h

Carl’sropeislongerthantheheightandshorterthanMartina’srope,sotherearetwopossibletriangles.Idrewthesecondtriangle.

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Situation 1:

36.0°

7.8 m5.9 m

MC

B

x

sin u7.8

5sin 36°

5.9

sin u 57.8 sin 36°

5.9 sin u 5 0.7770… u 5 sin−1 (0.7770…) u 5 50.9932…°

/B 5 180° 2 36.0° 2 50.9932…°/B 5 93.0067…°

x 2 5 5.92 1 7.82 2 2(5.9)(7.8) cos 93.0067…°x 2 5 100.4777… x 5 10.0238…In Situation 1, Martina and Carl are 10.0 m apart.

Situation 2:

36.0°

7.8 m5.9 m

MC y

B

sin u7.8

5 sin 36°

5.9

sin u 5 7.8 sin 36°

5.9 sin u 5 0.7770… u 5 sin−1 (0.7770…) u 5 50.9932…° u 5 180° 2 50.9932…° u 5 129.0067… °

/B 5 180° 2 36.0° 2 129.0067… °/B 5 14.9932…°

y 2 5 5.92 1 7.82 2 2(5.9)(7.8) cos 14.9932…°y 2 5 6.7433… y 5 2.5968…

Isubstitutedthesidelengthsandangles(includingu)intotheformulaforthesinelawandisolatedu.

Themeasuresoftheanglesinatrianglesumto180°.

Iusedthecosinelawtodeterminethedistance,x,betweenMartinaandCarl.Isubstitutedtheknownmeasurementsintothecosinelaw.

IalsoconsideredthesituationinwhichCarlisclosertoMartina.

Ideterminedthemeasureofthesupplementaryangle,whichissuitableforthissituation.

Iusedthesinelawtodetermineu.

Themeasuresoftheanglesinatrianglesumto180°.

Icanuse/Binthecosinelawtodeterminethedistance,y,betweenMartinaandCarl.

Isubstitutedthemeasureof/Bandthegivensidelengthsintothecosinelaw.

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In the second situation, Martina and Carl are 2.6 m apart.Martina and Carl are either 10.0 m apart or 2.6 m apart.

Your Turn

What length would Carl’s rope need to be in order for there to be only one possible triangle that could model this situation?

example 3 Reasoning about ambiguity

Leanne and Kerry are hiking in the mountains. They left Leanne’s car in the parking lot and walked northwest for 12.4 km to a campsite. Then they turned due south and walked another 7.0 km to a glacier lake. The weather was taking a turn for the worse, so they decided to plot a course directly back to the parking lot. Kerry remembered, from the map in the parking lot, that the angle between the path to the campsite and the path to the glacier lake measures about 30°. What compass direction should they follow to return directly to the parking lot?

Austin’s Solution

currentposition

S

N

W E

12.4 km

7.0 km

lake

campsite

parkinglot

30°

SinceIamgivenspecificdirections,Iknowexactlyhowtodrawasketchofthesituation.Thereisonlyonewaytodrawthesketch,sothisisnotambiguous.

LeanneandKerrylefttheparkinglotandwalkednorthwestandthensouth.

Becausethecampsiteisduenorthofthelake,Iknewthattheangleatthelakevertexofthetriangle,u,wouldhelpmedeterminethecompassdirectionthatLeanneandKerryneedtotravel.

MydiagramshowsthatLeanneandKerryneedtotravelapproximatelysoutheast.Pre-

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sin u12.4

5sin 30°

7.0

12.4a sin u12.4

b 5 12.4a sin 30°

7.0b

sin u 50.8857… u 5 sin−1(0.8857) u 5 62.3395…°

Correction:u 5 180° 2 62.3395…°u 5 117.6604…°

N

S

EW118°

S62°E

180° 2 117.6604…° 5 62.3395…°Leanne and Kerry would need to travel in the direction S62°E to reach the parking lot.

Your Turn

How far would Leanne and Kerry need to travel to reach the parking lot?

Isubtracted themeasureoftheangleinmytrianglefrom180°todeterminethedirectionoftravel.

Inoticedtwoside-anglepairs,soIsubstitutedthevaluesintothesinelawandsolvedforu.

Theangleseemedtoosmall,accordingtomydiagram.Tocorrectthe anglemeasure,Ineededthesupplementaryangle.

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In Summary

Key Idea

• The ambiguous case of the sine law may occur when you are given two side lengths and the measure of an angle that is opposite one of these sides. Depending on the measure of the given angle and the lengths of the given sides, you may need to construct and solve zero, one, or two triangles.

Need to Know

• In ^ ABC below, where h is the height of the triangle, / A and the lengths of sides a and b are given, and / A is acute, there are four possibilities to consider:

If / A is acute and a , h, there is no triangle.

A

b

a

h

If / A is acute and a 5 h, there is one right triangle.

A

b h a

If / A is acute and a . b or a 5 b, there is one triangle.

A

b ah

If / A is acute and h , a , b,there are two possible triangles.

A

b aa' h

• If / A, a, and b are given and / A is obtuse, there are two possibilities to consider:

If / A is obtuse and a , b or a 5 b, there is no triangle.

A

b

a

If / A is obtuse and a . b, there is one triangle.

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CHECK Your Understanding1. Given each set of measurements for ^ ABC, determine if there are zero,

one, or two possibilities. Draw the triangle(s) to support your answer. a) / A 5 75°, a 5 4 m, and b 5 12 m b) / A 5 50°, a 5 10 m, and b 5 6 m c) / A 5 115°, a 5 3.0 m, and b 5 9.0 m d) / A 5 62°, a 5 2.8 m, and b 5 3.0 m

2. Decide whether each description of a triangle involves the SSA situation.a) In ^ ABC, /B 5 100°, a 5 8 cm, and b 5 10 cm. b) In ^ DEF, /D 5 81°, e 5 9 cm, and f 5 8 cm. c) In ^GHI, /G 5 40°, i 5 5 cm, and g 5 4 cm. d) In ^ JKL, /L 5 15°, j 5 71 cm, and k 5 36 cm. e) In ^ MNO, /O 5 28°, m 5 8.4 cm, and o 5 4.0 cm. f ) In ^ PQR, /Q 5 95°, q 5 1.0 cm, and r 5 0.5 cm.

3. Calculate the height of each triangle in question 2. Determine the number of triangles that are possible (zero, one, or two). Justify your answers.

PRACTISING 4. Decide whether each description of a triangle involves the SSA

situation. If it does, determine the number of triangles (zero, one, or two) that are possible with the given measurements. Draw the triangle(s), and justify your answer. a) In ^ ABC, / A 5 51°, a 5 5 m, and b 5 14 m. b) In ^ ABC, /C 5 30°, a 5 6 mm, and c 5 12 mm. c) In ^ ABC, /B 5 40°, a 5 12 cm, and b 5 10 cm. d) In ^ ABC, / A 5 155°, b 5 15 m, and c 5 12 m.

5. In ^ DEF, EF 5 15.0 cm and /E 5 37°.a) Calculate the height of the triangle from

base ED.b) Determine the possible lengths of side FD, so that there are zero,

one, or two triangles that satisfy these conditions. Draw each triangle to support your answer.

6. A landowner claims that his property is triangular, with one side that is 430 m long and another side that is 110 m long. The angle that is opposite one of these sides measures 35°.a) Determine the length of the third side of the property, to the

nearest metre.b) Improve the description of the property to avoid confusion.

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7. The Raven’s Song, a traditional Tsimshian cedar canoe, is paddled away from a dock, directly toward a navigational buoy that is 5 km away. After reaching the buoy, the direction of the canoe is altered and it is paddled another 3 km. From the dock, the angle between the buoy and the canoe’s current position measures 12°.a) How far is the Raven’s Song from the dock?b) Is this the only possible solution? Explain.

8. An obtuse triangle has two known side lengths: 4.0 m and 4.2 m. The angle that is opposite the shorter side measures 64.0°.a) Calculate the obtuse angle in the triangle, to the nearest tenth of a

degree.b) Is there only one possible answer? Explain.

9. Part of a highway is to be cantilevered out from a mountainside, as shown. The width of the highway is 22 m, and the angle of the mountain slope at the road measures 51°. An 18 m beam needs to be installed to support the highway. Calculate possible distances, downhill from the highway, where the support post could be fastened. What distance would you recommend? Explain.

10. A farmer finishes repairing a fence post and then walks 250 yd through his corn field. He turns and walks another 300 yd east, until he can see the fence post southwest of him. He realizes that he left some of his tools at the fence post and heads directly back to it. How far does he need to walk, to the nearest metre?

11. In an extreme adventure triathalon, participants swim 1.7 km from a dock to one end of an island, run 1.5 km due north along the length of the island, and then kayak back to the dock. From the dock, the angle between the lines of sight to the ends of the island measures 15°. How long is the kayak leg of the race?

BillHelincarvedtheRaven’s Songfroma600-year-oldcedartakenfromtheNimpkishValley.ThecanoewascreatedtocarryamessageofgoodwillfromtheFirstNationsPeoplesoftheWestCoastofBritishColumbiatothe1994CommonwealthGamesinVictoria.


attachment pointfor support beam

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12. Carol is flying a kite on level ground. The string of the kite forms an angle of 50° with the ground. Two other girls, standing different distances from Carol, see the kite at angles of elevation of 66° and 35°. One girl is 11 m from Carol. All three girls are standing in a line. For each question below, state all possible answers to the nearest metre.a) How high is the kite above the ground?b) How long is the string?c) How far is the second girl from Carol?

13. The Huqiu Tower in China was built in 961 CE. When the tower was first built, its height was 47 m. Since then, it has tilted 2.8°. It is now called China’s Leaning Tower. There is a point on the ground where you can be equidistant from both the top and the bottom of the tower. How far is this point from the base of the tower? Round your answer to the nearest metre.

14. Create a SSA problem with zero, one, or two possible triangles. Exchange problems with a classmate. Sketch the situation described in your classmate’s problem, and determine the number of possible triangles.

15. Draw a SSA situation in which there is no possible triangle.a) Label the sides and angle, and use trigonometry to confirm that

there is no possible triangle.b) Determine the angle that would be necessary for there to be one

possible triangle.c) What angle would be necessary for there to be two possible

triangles?

Closing

16. In ^LMN, /L is acute. Using a sketch, explain the relationship among /L, sides l and m, and the height of ^LMN for each situation below.a) Only one triangle is possible.b) Two triangles are possible.c) No triangle is possible.

Extending

17. In ^DEF, d 5 13.0 cm, f 5 15.0 cm, and /D 5 26°. The two possible locations for vertex F are F1 and F2.a) Calculate the area of ^DEF1.b) Calculate the area of ^DEF2.c) Calculate the area of ^ F1EF2.d) Discuss with a classmate an alternative solution for determining

the area of ^ F1EF2.


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Dioptras and TheodolitesSurveying has played an important role in most cultures, including ancient cultures. For example, surveying tools and techniques were used to design the pyramids in ancient Egypt, map North America, and determine the boundaries of many nations. Tape measures, plumb lines, and levels were some of the original surveying tools. With the development of trigonometry, tools for measuring angles became important. One of these tools was the dioptra. It consisted of a sighting tube attached to a protractor, and it was used in ancient Greece and Rome to measure angles in the vertical and horizontal planes. Historians speculate that the Romans used dioptras as early as 2600 years ago, when building tunnels and aqueducts.

In the 16th century, more accurate tools were developed. A polimetrum, later called a theodolite, was used to measure horizontal angles. In the 18th century, the theodolite was combined with the altazimuth, an instrument for measuring vertical angles. The new, combined instrument became known as the modern theodolite or the transit theodolite.

A. Use a straw, a protractor, and a plumb line to construct your own dioptra.

45°0° string and weight

view throughhereprotractor

straw tube

45°

B. Use your dioptra and concepts of trigonometry to determine the height

of a building or a tree.

History Connection

YOU WILL NEED• protractor• string• straw• metre stick or tape measure

This instrument was modified to include a sighting telescope, which can measure angles to the nearest 2” or 0.06% of a degree.

Theodolites are still used in mapping and building. They can cost upward of $10 000.Pre-

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Analyzing an Area Puzzle

C

D

FA

B

S

E

^ ABC is an equilateral triangle with side lengths of 5 cm. Each side has been extended to the vertices of ^DEF. All the extended segments (CF, AD, and BE) are also 5 cm.

The Puzzle

A. Estimate how many ^ ABCs could fit into the area of ^DEF.

B. Using scissors and extra cutouts of ^ ABC, determine exactly how many ^ ABCs fit into ^DEF.

The Strategy

C. Describe the strategy you used to solve this puzzle.

Variation

D. Try using trigonometry to solve this puzzle.

E. Create a similar puzzle using a different regular polygon.

Applying Problem-Solving Strategies

YOU WILL NEED• scissors• calculator

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4.4 Solving Problems Using Obtuse Triangles

Solve problems that can be modelled by one or more obtuse triangles.

LEARN ABOUT the MathA surveyor in a helicopter would like to know the width of Garibaldi Lake in British Columbia. When the helicopter is hovering at 1610 m above the forest, the surveyor observes that the angles of depression to two points on opposite shores of the lake measure 45° and 82°. The helicopter and the two points are in the same vertical plane.

What is the width of Garibaldi Lake?

example 1 Visualizing a triangle to solve a problem

Determine the width of the lake, to the nearest metre.

Spencer’s Solution: Creating right triangles

lake

82°

45°

?

GOALYOU WILL NEED• calculator• ruler

Idrewadiagramofthehelicopterovertheforest,withitssightlines.

Anglesofdepressionarealwaysmeasuredagainstthehorizontal,soIdrewahorizontallineandplacedtheangles.

EXPLORE…

• Thecross-sectionofacanalhastwoslopesandistriangularinshape.Theanglesofinclinationfortheslopesmeasure28°and49°.Whenthecanalisfullofwater,thelengthofoneoftheslopesis12m.Whatisthewidthofthesurfaceofthewaterwhenthecanalisfull?

ThisviewofthesouthpartofGaribaldiLakewascapturedfromthePanoramaRidgetrail.

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NEL 1894.4 Solving Problems Using Obtuse Triangles

Lake

82°82°

45°

45°

Let a represent the distance from one end of the lake to the point directly below the helicopter.

Let b represent the distance from the other side of the lake to the point directly below the helicopter.

1610

45°

a

b

1610

82°

tan 45° 51610

a

a tan 45° 5 aa1610a b

a 51610

tan 45°a 5 1610

tan 82° 51610

b

b tan 82° 5 ba1610bb

b 51610

tan 82° b 5 226.270…

Width of lake 5 a 2 b Width of lake 51610 2 226.270…Width of lake 51383.730… m

The width of the lake is about 1384 m.

Becausethelakeisalsohorizontal,thealternateinterioranglesareequal.

Idrewthealtitudeofthehelicopteronthetriangle.Irealizedthatthesightlinesformtworighttriangles.

Iredreweachrighttriangle.

Forbothrighttriangles,themeasureofanangleandthelengthofitsoppositesideareknown.Theunknownbaseistheadjacentsideoftheangle.Iusedthetangentratiotodeterminethelengthofthebaseineachtriangle.

Sincea representsthewidthofthelakeandasmallpieceoflandbeneaththehelicopter,andbrepresentsthesmallpieceoflandbeneaththehelicopter, thewidthofthelakeisa2b.

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Emily’s Solution: Using the sine law

lake

37°45°

45° 98°

A

BC

D

1610

45°

A

B D

In ^ ABD:

sin 45° 51610AB

AB sin 45° 5 a1610AB

bAB

AB 51610

sin 45° AB 5 2276.883…

In ^ ABC:

AB

sin 98°5

BCsin 37°

sin 37°a2276.883...sin 98°

b 5 BC

1383.729… 5 BC

The width of the lake is about 1384 m.

Reflecting

A. Could Emily have used the cosine law to calculate the width of the lake?

B. Does Emily need to worry about the ambiguous case when using the sine law in this situation? Explain.

I drew a diagram to represent the situation.

I used parallel lines to determine the measure of / B. Then I calculated the remaining angle in the base to be 98°, since the measures of angles in a triangle add to 180°.

I calculated the angle at the helicopter, between the sight lines, by subtraction.

I used the primary trigonometric ratios to determine the length of AB. AB is a side in both ^ ABD and ^ ABC.

I used the sine law to determine the width of the lake, BC.

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APPLY the Mathexample 2 Solving a 3-D problem

A wind turbine called the Eye of the Wind is located at the top of Grouse Mountain in Vancouver. Rae is standing in the viewing pod at an altitude of 1272 m above sea level. She observes two ships in the harbour below. The first ship is at S3.3°E, with an angle of depression that measures 6.9°. The second ship is at S15.5°E, with an angle of depression that measures 7.3°. Determine the distance between the two ships, to the nearest metre.

Rae’s Solution

ship 1ship 2

1272 mE

F

GH

Let a represent the horizontal distance from Rae to ship 1.

Let b represent the horizontal distance from Rae to ship 2.

1272 m

6.9°83.1°

a ship 1

E

G H

In ^ EGH:

tan 83.1° 5a

1272

1272 tan 83.1° 5 a a1272

b 1272

10 511.2416… 5 a

The Eye of the Wind was built in 2009. The power that it generates is about 20% of the total power required for Grouse Mountain. There is an elevator up to the viewing pod, where visitors can see Vancouver and the surrounding mountains.

The angle between the altitude of the viewing platform and the horizontal measures 90°. If the angle of depression measures 6.9°, then the measure of the complementary angle in the triangle is 83.1° because these measures must add to 90°.

I sketched a 3-D diagram of this situation. I noticed that there are two right triangles.

I decided to draw the right triangles separately.

These are right triangles, so I used the tangent ratio to determine the horizontal distance from the base of the mountain, a and b, to each ship.

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1272 m

82.7°

7.3°

bship 2

E

GF

In ^ EGF:

tan 82.7° 5b

1272

1272 tan 82.7° 5 a b1272

b 1272

9929.5133… 5 b

ship 2

ship 1S

3.3°12.2°

15.5° 9930 m

10 511 m

x

N

W EG

F

H

x 2 5 (9930)2 1 (10 511)2 2 2(9930)(10 511) cos 12.2°x 2 5 5 052 701.96 x 5 2247.8216…

The distance between the two ships is about 2248 m.

Your Turn

If you were on the bridge of ship 2, in what direction would ship 1 be?

Idrewthesituation,asseenfromabovethewindturbine.

Bothcompassdirectionsaremeasuredagainstsouth,soIdrew anorth–southlineandtheapproximatesightlinestoeachship.

Todeterminethemeasureoftheanglebetweenthetwosightlines,Isubtracted:

15.5°23.3°512.2°

Thisdistanceappearsappropriate,accordingtomydiagram.

InoticedthatIhadtwoknownsidesandacontainedangle,soIusedthecosinelawtodeterminethedistancebetweenthetwoships.

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In Summary

Key Idea

• Thesinelaw,thecosinelaw,theprimarytrigonometricratios,andthesumofthemeasuresoftheanglesinatrianglemayallbeusefulwhensolvingproblemsthatcanbemodelledusingobtusetriangles.

Need to Know

• Whensolvingproblemsthatinvolvetrigonometry,thefollowingdecisiontreemaybeusefulforchoosinganappropriatestrategy.

Draw and label a diagram withall the given information.

Righttriangle

Acutetriangle

Obtusetriangle

Use thesine law* or cosine law.

* When you know the lengths of two sides and the measure of an angle that is not contained by the two sides, the case may be ambiguous.

Use theprimary

trigonometricratios.

CHECK Your Understanding1. a) Explain how you would determine the indicated side length or

angle measure in each triangle. i) ii) iii)

18 m30°

14° x15 cm

8 cm

20°

1.3 m

1.0 m

0.9 m

b) Use the strategies you described to determine the indicated side lengths and angle measure in part a). Round your answers to the nearest tenth of a unit.

c) Compare your strategies with a classmate’s strategies. Which strategy seems to be more efficient for each triangle?

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PRACTISING2. Two forest-fire towers, A and B, are 20.3 km apart. From tower A, the

compass heading for tower B is S80°E. The ranger in each tower sees the same forest fire. The heading of the fire from tower A is N50°E. The heading of the fire from tower B is N60°W. How far, to the nearest tenth of a kilometre, is the fire from each tower?

tower A

tower B

fire

20.3 km

N

N

60°

50°

3. The Leaning Tower of Pisa is 55.9 m tall and leans 5.5° from the vertical. What is the distance from the top of the tower to the tip of its shadow, when its shadow is 90.0 m long? (Assume that the ground around the tower is level.) Round your answer to the nearest metre.

4. Shannon wants to build a regular pentagonal sun deck. She is going to use five 2-by-6s, each 12 ft long, to frame the perimeter. She plans to finish the deck with 4 in. cedar planks, laid side by side and parallel to one of the sides. Determine the length of the longest cedar plank.

5. Bijan is hiking in Manning Park, British Columbia. He is hiking alone, but he has a walkie-talkie so that he can keep in touch with his friends at the camp. The walkie-talkies have a range of 6 km. Bijan hikes 5 km along the Skagit Bluffs Trail in a S60°E direction. He then hikes 2 km along the Hope Pass Trail in a N30°E direction.a) Draw a diagram to show Bijan’s hiking route. Estimate his distance

from the camp. Is he still in the range to communicate with his friends at the camp?

b) Calculate Bijan’s distance from the camp. Can he still communicate with his friends at the camp? Explain.

55.9 m

5.5°

shadow

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6. On February 28, 2010, Earth was equidistant from the spacecraft Dawn and the Sun, forming an isosceles triangle. The distance from Earth to Dawn and Earth to the Sun was 0.99 AU (astronomical units). The distance from Dawn to the Sun was 1.84 AU.a) Draw a diagram to show Dawn, Earth, and the Sun.b) Determine the angle between the sight lines from Earth to Dawn

and the Sun.

Dawn was launched by NASA on September 27, 2007, with the goal of investigating two of the largest objects in the main asteroid belt: Vesta and Ceres. Dawn was to arrive at Vesta in July 2011 and at Ceres in February 2015.

7. A surveyor is measuring the length of a lake. He takes angle measurements from two positions, A and B, that are 136 m apart and on the same side of the lake. From B, the measure of the angle between the sight lines to the ends of the lake is 130°, and the measure of the angle between the sight lines to A and one end of the lake is 120°. From A, the measure of the angle between the sight lines to the ends of the lake is 65°, and the measure of the angle between the sight lines to B and the same end of the lake is 20°. Calculate the length of the lake, to the nearest metre.

8. From an airplane, the angles of depression to two forest fires measure 18° and 35°. One fire is on a heading of N15°W. The other fire is on a heading of S70°E. The airplane is flying at an altitude of 3000 ft. What is the distance between the two fires, to the nearest foot?

9. Bert wants to calculate the height of a tree on the opposite bank of a river. To do this, he lays out a baseline that is 80 m long and measures the angles shown in the diagram. Is the information that Bert has gathered sufficient to determine the height of the tree? Justify your answer.

A

30°

80 m85°

28°

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10. Two towns, Smith Falls and Chester, are 20 km apart. From Smith Falls, the direction to Chester is N70°E. A grass fire has been reported on a bearing of N30°E from Smith Falls and N12°E from Chester. Which town’s fire department is closer to the fire? How much closer is it, to the nearest kilometre?

11. Mount Logan, in Yukon Territory, is Canada’s highest peak. In North America, it is second in height only to Mount McKinley. An amateur climber is trying to calculate the height of Mount Logan. From her campsite, the angle of elevation to the summit measures 35°. She walks 500 m closer, up a 10° inclined slope, and measures the new angle of elevation as 38°. Her campsite is at an altitude of 1834 m. Determine the height of Mount Logan, to the nearest 10 m.

12. Brit and Tara are standing 8.8 m apart on a dock when they observe a sailboat moving parallel to the dock. When the sailboat is equidistant from both girls, the angle of elevation to the top of its 8.0 m mast is 51° for both girls. Describe how you would determine the measure of the angle, to the nearest degree, between Tara and the boat, as viewed from Brit’s position. Justify your answer.

13. A zip line is going to be suspended between two trees. From the forest floor, 12 m from the base of the smaller tree, the angles of elevation to the tree platforms measure 33° and 35°. The distance between the two trees is 35 m.a) Draw a diagram to represent this situation. What assumptions did

you make?b) Calculate the length of the zip line needed.

14. Determine the angle of depression for the zip line in question 13.

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Closing

15. Sketch an obtuse oblique triangle, and label any three measurements (side lengths or angles). Exchange triangles with a classmate. Solve your classmate’s triangle, if possible. If your classmate’s triangle is impossible to solve, explain to your classmate why it is impossible to solve.

Extending

16. A sailor, out on a lake, sees two lighthouses that are 11 km apart. Lighthouse A is in the direction N47°W and lighthouse B is in the direction N5°W. As seen from lighthouse A, lighthouse B is in the direction N8°E.a) How far, to the nearest kilometre, is the sailor from each

lighthouse?b) Assuming that the shore runs on a straight line through both

lighthouses, what is the least distance from the sailor to the shore? Round your answer to the nearest kilometre.

17. An airport radar operator locates two airplanes that are flying toward the airport. The first airplane, P, is 120 km from the airport, A, in a N70°E direction and at an altitude of 2.7 km. The other airplane, Q, is 180 km away, in a S40°W direction and at an altitude of 1.8 km. Calculate the distance between the two airplanes to the nearest tenth of a kilometre.

Math in Action

Measuring the Viewing Angle of a Screen

Televisionsets,computermonitors,andotherscreensareoftenrankedaccordingtotheirviewingangle.Ifascreenhasaviewingangleof80°,thismeansthatthecolour,brightness,andimagequalitydonotappeartodegradeuntilyouareviewingthescreenatanangleof80°ormore,measuredfromthecentralaxisofthescreen.•Withapartnerorinasmallgroup,makeaplantomeasuretheviewingangleofascreen.

•Testyourplan.Whatadjustmentsdidyouneedtomakeasyoumeasured?

•Compareyourresultswiththeresultsofotherpairsorgroups.Areyousatisfiedthatyourplanworkedwell?Explain.

viewingangle

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41. Determine a, where 0 < a < 180°, for each trigonometric ratio below.

Round your answer to the nearest tenth of a degree.

a) cos a 5 0.235 c) sin a 5 0.015 e) sin a 512

b) tan a 5 2.314 d) cos a 5 234

f) sin a 5 0.600

2. For each description below, determine if there are zero, one, or two possible triangles. Draw the triangle(s), if possible, including the unknown measurements.a) In ^DEF, d 5 5 cm, e 5 3 cm, and f 5 9 cm.b) In ^ ABC, / A 5 25°, b 5 3 m, and c 5 10 m.c) In ^ JKL, / J 5 55°, j 5 10.4 km, and k 5 11.6 km.d) In ^ PQR, / P 5 17°, /Q 5 110°, and r 5 26 mm.e) In ^ FUN, / F 5 75°, f 5 25 cm, and n 5 47 cm.

3. Two workers are helping a crane operator lower a crate to the ground. The workers are standing in line with the crate and each other. Each of them has a rope attached to the crate. The first worker has a 35 ft rope that makes an angle of 50° with the ground. The second worker has a 30 ft rope. Determine the distance, to the nearest foot, between the two workers.

4. A vertical tree stands on the side of a ski run. The angle of inclination for the ski run measures 40°. When the angle of elevation of the Sun measures 65°, the tree casts a shadow 75.0 m down the slope. Determine the height of the tree, to the nearest tenth of a metre.

5. ^ ABC is an equilateral triangle with a perimeter of 36 cm. Three triangles are created when / A is divided into three equal angles. Two of these triangles are obtuse. Determine the side lengths of the obtuse triangles, to the nearest centimetre.

6. The following describes the location of a buried treasure. From the pine tree, walk 30 paces N20°E, then turn and walk 15 paces until the tree is due south. How many paces would you need to walk due north of the tree to reach the buried treasure?

7. From the window of a building, 45 m up, the angles of depression to two different intersections measure 76° and 65°. The measure of the angle between the lines of sight to the two intersections is 135°. Calculate, to the nearest metre, the distance between the two intersections.

WHAT DO You Think Now? Revisit What Do You Think? on page 161. How have your answers and explanations changed?

Chapter Self-Test

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NEL 199Chapter Review

4FREQUENTLY ASKED QuestionsQ: In a SSA situation, how do you know how many triangles

are possible?

A: In a SSA situation, you know the lengths of two sides and the measure of an angle that is opposite one of the sides. After drawing a diagram for the situation, you should first determine the height of the triangle, opposite the known angle. Then you can determine the number of possible triangles by comparing the height of the triangle with the length of the side that is opposite the given angle.• Iftheheightisgreaterthanthelengthofthesidethatisopposite

the given angle, no triangle is possible.• Iftheheightequalsthelengthofthesidethatisoppositethegiven

angle, one triangle is possible.• Iftheheightislessthanthelengthofthesidethatisoppositethe

given angle, two triangles may be possible: - If the length of the side that is opposite the given angle is less

than the length of the other given side, two triangles are possible. - If the length of the side that is opposite the given angle is greater

thanorequaltothelengthoftheothergivenside,onetriangleispossible.

Q: When solving a problem that can be modelled by an obtuse triangle, how do you decide whether to use the sine law or the cosine law?

A: Use the same decision process that you used for acute triangles:• Drawaclearlylabelleddiagram.Includegiveninformationand

information you can deduce.• Usethesinelawifyouhaveeitherofthefollowingsituations:

A

Cb

A

b a

• Usethecosinelawifyouhaveeitherofthesesituations:

cc

b

A

ba

Study Aid•SeeLesson4.3,Examples1to3.

•TryChapterReviewQuestions5and7.

Study Aid•SeeLesson4.4,Examples1and2.

•TryChapterReviewQuestion8.

Chapter Review

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PRACTISINGLesson 4.1

1. Describe, using examples, the relationships between the primary trigonometric ratios for supplementary angles.

2. Determine each trigonometric ratio. Predict another angle that has an equal or opposite ratio. Check your prediction.a) sin 122° c) cos 100°b) sin 58° d) tan 15°

Lesson 4.2

3. Determine the unknown angle measure that is indicated in each triangle, to the nearest tenth of a unit.a)

b)

4. In ^ ABC, / A 5 125°, /B 5 30°, and the side between these angles is 8.0 cm long. Solve the triangle. Round each measure to the nearest tenth of a unit, as necessary.

Lesson 4.3

5. For each description, determine the number of possible triangles. Draw the triangle(s) to support your answer.a) In ^ ABC, / A 5 53°, a 5 7 m, and

b 5 15 m.b) In ^ ABC, / A 5 27°, a 5 5 m, and

b 5 6 m.c) In ^ ABC, / A 5115°, a 5 23.0 m, and

b 5 6.0 m.

6. Determine the unknown side length or angle measure that is indicated in each triangle, to the nearest tenth of a unit.a)

b)

7. A 4.3 m ramp for a mountain-bike trail is inclined at a 15° angle with the ground. The length of the support that creates the incline is 1.3 m.a) Determine the distance along the ground

between the base of the support and the beginning of the ramp, to the nearest tenth of a metre.

b) How high above the ground is the take-off point on the ramp, to the nearest tenth of a metre?

Lesson 4.4

8. An airplane passes over an airport and continues flying on a heading of N70°W for 3 km. The airplane then turns left and flies another 2 km until the airport is exactly due east of its position. What is the distance between the airplane and the airport, to the nearest tenth of a kilometre?

4.8 cm5.3 cm

118°

21.2 cm 16.5 cm

16.2 cm

5.0 m

4.0 m

42°

x

8.0 m9.8 m

32°

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NEL 201Chapter Task

4Stage Lighting and Trigonometry

Whether you are at a rock concert or a play, lighting has an important role in making the show enjoyable. When designing for a stage, the lighting designer must consider the focus areas, which are the areas where the director wants the audience to look. Each focus area may require light from multiple sources.

Suppose that you are designing the lighting for a presentation. You are using four automated lights hung from a V-shaped bar in the ceiling, directly above the stage. The presentation requires two main focus areas on the stage, spaced 5 m apart, as shown in the diagram.

Lights 1 and 2 have already been set and measured to focus on area A. The measures of the angles are 70° and 55°, with respect to the bar. You now need to program all four lights so that they can be directed to each focus area. Each light must be programmed with an angle measure from the vertical, not from the bar.

What angle measures are needed for each light to illuminate each focus area?

A. Determine the angle measures from the vertical so that lights 1 and 2 will shine on focus area A.

B. Determine the angle measures from the bar so that lights 3 and 4 will shine on focus area A.

C. Determine the angle measures from the bar so that lights 1 to 4 will shine on focus area B.

D. Determine the angle measures from the vertical to program all four lights so they can be directed to shine on both focus areas.

?

Chapter Task

Task Checklist✔ Didyoudrawclear,labelled

diagrams?

✔ Didyouexplainyoursolutionclearly?

light 1

stage left stage right

light 2 light 3

light 4

A B

5 m

150°

1 m

2 m

1 m

2 m

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4Carrying Out Your ResearchAs you continue with your project, you will need to conduct research and collect data. The strategies that follow will help you collect data.

Considering the Type of Data You Need

There are two different types of data that you need to consider: primary and secondary. Primary data is data that you collect yourself using surveys, interviews, and direct observations. Secondary data is data you obtain through other sources, such as online publications, journals, magazines, and newspapers.

Both primary data and secondary data have their pros and cons. Primary data provides specific information about your research question or statement, but may take time to collect and process. Secondary data is usually easier to obtain and can be analyzed in less time. However, because the data was gathered for other purposes, you may need to sift through it to find what you are looking for.

The type of data you choose can depend on many factors, including the research question, your skills, and available time and resources. Based on these and other factors, you may choose to use primary data, secondary data, or both.

Assessing the Reliability of Sources

When collecting primary data, you must ensure the following:• Forsurveys,thesamplesizemustbereasonablylargeandtherandom

sampling technique must be well designed.• Forsurveysorinterviews,thequestionnairesmustbedesignedto

avoid bias.• Forexperimentsorstudies,thedatamustbefreefrommeasurementbias.• Thedatamustbecompiledaccurately.

Project Connection

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NEL 203Project Connection

When obtaining secondary data, you must ensure that the source of your data is reliable:• Ifthedataisfromareport,determinewhattheauthor’scredentialsare,

how up-to-date the data is, and whether other researchers have cited the same data.

• Beawarethatdatacollectionisoftenfundedbyanorganizationwithan interest in the outcome or with an agenda that it is trying to further. When presenting the data, the authors may give higher priority to the interests of the organization than to the public interest. Knowing which organization has funded the data collection may help you decide how reliable the data is, or what type of bias may have influenced the collection or presentation of the data.

• IfthedataisfromtheInternet,checkitagainstthefollowingcriteria: - Authority: The credentials of the author are provided and can be

checked. Ideally, there should be a way to contact the author with questions.

- Accuracy: The domain of the web address may help you determine the accuracy of the data. For example, web documents from academic sources (domain .edu), non-profit organizations and associations (domains .org and .net) and government departments (domains such as .gov and .ca) may have undergone vetting for accuracy before being published on the Internet.

- Currency: When pages on a site are updated regularly and links are valid, the information is probably being actively managed. This could mean that the data is being checked and revised appropriately.

Accessing Resources

To gather secondary data, explore a variety of resources:• textbooks• scientificandhistoricaljournalsandotherexpertpublications• newsgroupsanddiscussiongroups• librarydatabases,suchasElectricLibraryCanada,whichisadatabaseof

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People may be willing to help you with your research, perhaps by providing information they have or by pointing you to sources of information. Your school or community librarian can help you locate relevant sources, as can the librarians of local community colleges or universities. Other people, such as teachers, your parents or guardians, local professionals, and Elders and Knowledge Keepers may have valuable input. (Be sure to respect local community protocols when approaching Elders or Knowledge Keepers.) The only way to find out if someone can and will help is to ask. Make a list of people who might be able to help you obtain the information you need, and then identify how you might contact each person on your list.

project example Carrying out your research

Sarah chose, “Which Western province or territory grew the fastest over the last century, and why?” as her research question. She has decided to use 1900 to 2000 as the time period. How can she find relevant data?

Sarah’s Search

Sincemyquestioninvolvesahistoricaleventoverawidearea,Idecidedtorelyonsecondarydata.IstartedmysearchusingtheInternet.Ididasearchfor“provincialpopulationsCanada1900to2000”andfound manywebsites.IhadtolookatquiteafewuntilIfoundthefollowinglink:

ThisledmetoadocumentfromtheUniversityofBritishColumbia that citedadocumentfromStatisticsCanada,basedoncensusdata,thatshowedtheprovincialpopulationsfrom1851to1976.IwenttotheStatisticsCanadawebsiteandsearchedforthecensusdata,butIcouldn’tfindit.SoItriedanothergeneralsearch,“historicalstatisticsCanadapopulation,”andfoundthelinkbelow:


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ThisledmetodataIwaslookingfor:

InowhavesomedataIcanuse.Ifeelconfidentthatthedataisauthoritativeandaccurate,becauseIbelievethatthesourceisreliable.Iwillcontinuelookingformorecurrentdata,from1976to2000.Iwillalsoneedtosearchforinformationaboutreasonsforpopulationchangesduringthistimeperiod.Iwillasktheschoollibrariantohelpmelookforothersources.

NEL 205Project Connection

Your Turn

A. Decide if you will use primary data, secondary data, or both. Explain how you made your decision.

B. Make a plan you can follow to collect your data.

C. Carry out your plan to collect your data. Make sure that you record your successful searches, so you can easily access these sources at a later time. You should also record detailed information about your sources, so you can cite them in your report. See your teacher for the preferred format for endnotes, footnotes, or in-text citations.

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Documents

Pre-Publication110˚ · 30 GPS satellites. Also in 2010, Russia had almost completed its own system of 24 satellites, and both the European Community and China were beginning to deploy