Tran is a park warden. She walked 5.5 km west from A along a straight trail. Then she turned at B and walked 4.5 km southeast to C. How far was Tran from where she started?

1 Can you use the sine law to solve this problem? Explain.

2 You can use the cosinelaw to solve this problem. Two forms of the cosine law are shown below.

a2 5 b2 1 c2 2 2bc cos A b2 5 a2 1 c2 2 2ac cos B

Each form uses the cosine of the angle that is opposite the side whose length you want to calculate. Look at the pattern. What is the third form of the cosine law?

c2 5 2 1 2 2 2 cos

3 How far was Tran from point A, where she started?

b2 5 a2 1 c2 2 2ac cos B

b2 5 2 1 2 2 2( )( )cos 8

b2 5 1 2 cos 8

b2 5

b 5 Tran was km from where she started.

Example Tijana makes pendants from stained glass. She sells the pendants at craft shows. She needs a triangular piece of glass with the dimensions shown. At what angle should Tijana cut the glass at P? Record the answer on the diagram.

Exploring the cosine Law 9 .4Try These

i) x2 5 32 1 42 2 (0.5)(14)(3) ii) x2 5 5.32 1 2.72 2 (2)(5.3)(2.7)

x 5 x 5

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76°3.5 cm

P

S

Q 5.0 cm

4.5 cm

42 1 52 2 2(4)(5)cos 308 To calculate in one step, enter4 x2 1 5 x2 2 2 3 4 3 5 3 cos 30 5

Tech Tip

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AB

C

45°

b

5.5 km

4.5 km

N

W

S

E

3.9 km

NW

SW SE

NE

REfLEcTinGWhy does it make sense to use the

cosine law to calculate b2?

cosinelawfor nABC, the cosine law is written asa2 5 b2 1 c2 2 2bc cos A

2

a

4.5

20.25

15.498…

3.936…

30.25 49.5

45

45

3.9

4.55.5 5.5

b ab C

No. e.g., You would need to know one side

and the opposite angle to use the sine law.

2.6

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Solution1:UsingthecosinelawA. What are the side lengths?

p 5 cm q 5 cm s 5 cm

B. At what angle should Tijana cut the glass at P?

p2 5 q2 1 s2 2 2qs cos P 2 5 2 1 2 2 2( )( ) cos P

5 1 2 cos P

2 2 5 cos P

5 cos P

5 cos P

cos-1 a b 5 /P

8 5 /P

Tijana should cut the glass at an angle of 8.

Solution2:RearrangingthecosinelawA. Substitute the values for the three sides into the rearranged

cosine law. Solve for /P.

cos P 5 q2 1 s2 2 p2

2qs

cos P 5 2 1 2 2 2

21 2 1 2

cos P 5

  /P 5 cos–1 a b

/P 5 8, or 8

Use brackets around both the numerator and the denominator. To

calculate 52 1 82 2 62

2152 182 , enter

( 5 x2 1 8 x2 2 6 x2 ) 4 ( 2 3 5 3 8 ) 5

Tech Tip

RearrangingthecosinelawTo determine an angle, you can write the cosine law like this:

cos A 5 b2 1 c2 2 a2

2bc

cos B 5 a2 1 c2 2 b2

2ac

cos C 5 a2 1 b2 2 c2

2ab

Hint

5.0 3.5 4.5

3.5

12.25

12.25

5.0

25

25

76.225…

27.5 231.5

76

231.5

31.5

3.5

31.5

231.5

4.5

20.25

20.25

4.5

27.5

7.5

3.5 4.5 5.0

4.53.5

31.5

31.5

7.5

7.5

76.225… 76

Chapter 9 Trigonometry 231NEL

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Practice 1. Write the form of the cosine law needed to determine each

measure.

a) /N 

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S

n x

sX N

b) side ED 

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Q

d e

qE D

2. What is the measure of the unknown side length in each triangle?

a)

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D

20 in. 78° 12 in.

21 in.F G

b)

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23°11.6 cm

10.8 cm4.5 cm

C

K T

3. What is the measure of each angle?

a) /H 

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37°45 cm

39 cm 27 cm

H M

N

b) /C 

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76°

44 in.

39 in.

33 in.

C

F

X

c2 5 f 2 1 x2 2 2fx cos C

REfLEcTinGWhich method for determining the

angle measure in Question 3 did

you prefer? Why?

cos H 5 m2 1 n2 2 h2

2mn

REfLEcTinGHow can you write the cosine law in

Question 1, Part a) to determine /S?

 d2 5 f 2 1 g2 2 2fg cos D

n2 5 s2 1 x2 2 2sx cos NOR cosN5s2 1 x2 2 n2

2sxq2 5 d2 1 e2 2 2de cos Q

d2 5 122 1 202 2 2(12)(20) cos 788

d2 5 444.202…

d 5 21.076…, or 21 in.

e.g., t2 5 k2 1 c2 2 2kc cos T

t2 5 10.82 1 11.62 2 2(10.8)(11.6) cos 238

t2 5 20.558…

t 5 4.534…, or 4.5 cm

cos H 5 392 1 452 2 272

2(39)(45)

cos H 5 2817

3510

/H 5 cos21a 2817

3510b

/H 5 36.624…8, or 378

442 5 332 1 392 22(33)(39) cos C

1936 5 1089 1 1521 2 2754 cos C

1936 2 1089 2 1521 5 2 2754 cos C

2674

22754 5 cos C

cos21a 2674

22754b 5 /C

75.833…8 5 /C, or 768

232 Apprenticeship and Workplace 12 NEL

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4. a) Determine the length of side c using each method.

Cosine law

c2 5 a2 1 b2 2 2ab cos C

Pythagorean theorem

c2 5 a2 1 b2

b) What is the value of 2ab cos C if /C is a right angle?

c) Simplify the cosine law if /C is a right angle.

c2 5 a2 1 b2 2 2ab cos 908

What is the result?

5. Dominique is a window dresser for a department store. This diagram shows how she plans to use a brace to support a collage that is 40 in. tall. How long does the brace need to be?

6. The Louvre Pyramid, in Paris, is at the main entrance to the Louvre Museum. Its dimensions are shown on the photo at the right. What is the measure of each angle between the base and the side edges?

7. Jaani is a contractor in Hanna. How might Jaani use the cosine law when he is building a roof?

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8 ft

15 ft

A

C B

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82°

41 in.

40 in.(art)

16 in

a(brace)

A C

B

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35.42 m

33.10 m

A

B C

33.10 m

c2 5 152 1 82

c2 5 289

c 5 17, or 17 ft

e.g., The cosine law becomes c2 5 a2 1 b2.

This is the Pythagorean theorem.

e.g., If Jaani knows the length of each rafter and the width of the structure,

he can use the cosine law to calculate the angles.

e.g., a2 5 b2 1 c2 2 2bc cos A

a2 5 162 1 402 2 2(16)(40) cos 828

a2 5 1677.858…

a 5 40.961… The brace needs to be 41 in. long.

e.g., cos B 5 a2 1 c2 2 b2

2ac

cos B 5 35.422 1 33.102 2 33.102

2(35.42)(33.10)

  /B 5 cos–1a1254.5764

2344.804b , or 57.653…

Each angle between the base and the side edges is 588.

c2 5 152 1 82 2 2(15)(8) cos 908

c2 5 289

c 5 17, or 17 ft

0

Chapter 9 Trigonometry 233NEL

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