Law of Cosines 8 6

[Pages:6]86 Law of Cosines

Objective To apply the Law ofCosines

Mathematics Florida Standards

MAFS.912.G-SRT.4.11 Understand and apply

the... Law of Cosines... Also MAFS.912.G-SRT.4.10 MP 1, MR 3, MP 4, MP 7

Getting Ready!

OX C

In the diagram. AABC Is an acute triangle.

Think obout how the areas of

the squares are

Use what you know about right triangle trigonometry to write an expression for the area of the shaded region that uses

related to the

a, b. and C.

side lengths of

the triangle. What

does this remind

you of?

J-

PRA^1c"s Solve It, you used right triangle trigonometry to write an expression to describe

a side length.You can also find relationships between the angle measures and the side lengths of nonright triangles.

Essential Understanding Ifyou knowthe measuresoftwo side lengthsand the

measure ofthe included angle(SAS),or allthree side lengths(SSS),then you can find all the other measures ofthe triangle.

Lesson

Vocabulary

Law of Cosines

Key Concept Law of Cosines

For any AABC,the Law of Cosines relates the cosine ofeach angle to the side lengths ofthe triangle. ,2- = + ^2-2bccos A

= +(^ -- 2accos B (^ = + b^ -- 2ab cos C

Here's Why it Works Note that

and x = b cos A.

Use the Pythagorean Theorem with ABCD and simplify.

=(c-x)^ +

Pythagorean Theorem

a^ = c^ --2cx4-x^ + b^ Simplify.

a^ = c^ - 2cb cosA + b^ Substitute for

and b cos A for x.

a^=b^ +c^-IbccosA Commutative Property

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Lesson 8-6 Law of Cosines

527

Problem 1 Using the Law of Cosines(SAS)

Find b to the nearest tenth.

. A

AB is opposite Z.C

so -4S = c = 10.

BC is opposite ZA.

so SC = a = 22. = 44

the length of;4C

Because you know mz.B and need b,substitute the angle measure

and the two side lengths into =a^ +c^ - 2accosfi and

solve for b.

-2accosB

Law of Cosines

= 22^ + 10^ - 2(22)(10) cos 44? Substitute.

b = 16.35513644

Use a calculator.

The value of b is about 16.4.

Got It? 1. Find MN to the nearest tenth.

04?/29

How can you use what you know to

find mz.V? You know the three side

lengths(SSS)so you can

use the Law of Cosines to find m/.V.

Problem 2 Using the Low of Cosines(SSS)

In ATUV,TU= 4.4, UV=7.1,and TV= 6.7.Find mLV to the nearest tenth ofa degree.

Step 1 Draw and label a diagram.

Step 2 Use the Law ofCosines to set up an equation.

TU^=(/y2+ 7V2-2{UV){TV)cos V

4.42= 7.1^+6.7^-2(7.1)(6.7)cos V

Step 3 Solve for mZ.V.

19.36 = 50.41 + 44.89-95.14 cos

-75.94 = cos V -95.14

Simplify.

Solve for cos V.

Solve for mZ-U".

37.04219062

Use a cadulator.

The measure of Z.y is about 37.0.

^GoUt? 2.In ATC/Vabove,find mATtothenearesttenthdegree.

528 Chapter 8 Right Triangles and Trigonometry

You can use the Law of Cosines to solve real world problems involving triangles.

What do you need to find before you

can use the law of Cosines? You need to find the

measure of the angle opposite d in the triangle before you can apply the

Law of Cosines,

Problem 3 Using the Law of Cosines to Solve a Problem

An air traffic controller is tracking a plane 2.1 kilometers due south ofthe radar tower.A second plane is located 3.5 kilometersfrom the tower at a heading ofN 75? E (75? east ofnorth).To the nearest tenth ofa kilometer, how far apart are the two planes?

North

Plane B 3.5 km

Tower

The north-south line in the figure represents a straight angle. Let the angle opposite d be Z.D.Use supplementary angles to find mAD.Use supplementary angles to find the measure ofthe angle opposite d.

m/LD = 180 -- 75 = 105

Supplementary angles

2.1 km Plane A

Use the Law of Cosines to solve for d.

d^~a^ + b^-lab cosD

Law of Cosines

= 3.52 -H 2.]^- 2{3.5)(2.1) cos 105? Substitute.

d 4.52378602

Use a calculator.

The distance between the two planes is about 4.5 kilometers.

^Got It? 3. You and afriend hike 1.4 milesdue westfrom a campsite.Atthesametime,

two other friends hike 1.9 miles at a heading ofS 1r W(11? west ofsouth) from the campsite.To the nearest tenth ofa mile,how far apartare the two groups?

Lesson Check Do you know HOW?

1. In AABC,AB = 7,BC= 10,and mAB = 80. To the nearest tenth, what is b?

2. In AQRS,QR = 31.9, RS= 25.2,and QS= 37.6. To the nearest tenth, what is mARl

3. In ALMN,LN=7,MN= 10,and mAN= 48. To the nearest tenth, what is the area of ALMNl

4. What are mAX,mAY, and mAZl

Do you UNDERSTAND?

MATHEMATICAL

PRACTICES

5. Error Analysis In AABC,AC= 15ft, BC= 12ft, and mAC=32.A student solved for c for a = 12ft, b = 15ft, and mAC= 32.What was the error?

C= 369-

6. Reasoning Explain how you would find the measure ofthe largest angle ofa triangle ifgiven the measures ofthe three side lengths.

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Lesson 8-6 Law of Cosines

529

Practice and Problem-Solving Exercises

MATHEMATICAL

PRACTICES

Practice

Use the information given to solve.

^ See Problems 1 and 2.

7. In AQRS,mAR =38, QR= 11,and fiS= I6.T0 the nearest tenth,whatis the length of QS?

8. In AWXY, WX= 20.4, XY= 16.4,and WY= 25.3.To the nearesttenth,what is

mAWl

9. In AJKL,}K -- 2.6, KL = 6.4, and mAK= 10.5.To the nearest tenth, whatis the length of JL?

10. In ADEF,DE = 13,EE= 24,and FD -- 27.To the nearest tenth, what is mAEl

Use the Law ofCosines to find the values ofjc andy.Round to the nearest tenth.

13.

Use the Law of Cosines to solve each problem.

15. Baseball After fielding a ground ball,a pitcher

is located 110 feet from first base and 57 feet from

home plate as shown in the figure at the right. To the nearesttenth, whatis the measure ofthe angle with its vertex at the pitcher?

16. Zipllne One side ofa ravine is 14ft long.The other side is 12ft long.A 20ft zipline runs from the top ofone side

ofthe ravine to the other.To the nearest tenth, at what angle do the sides ofthe ravine meet?

20 ft

^ See Problem 3.

pitcher

57 ft Home Plate

Apply

14ft

17. Think About a Plan A walking path around the outside ofa garden is shaped like a triangle.Two sides ofthe path that measure 32ft and 39 ftform a 76? angle.Ifyou walk around the entire path one time,how far have you walked? Write your answer

to the nearest foot.

? What information do you need to find before you can solve this problem? ? How can you find the information you need? ? Can drawing a diagram help you solve this problem?

530 Chapter 8 Right Triangles and Trigonometry

18. Airplane A commuter plane flies from City A to City B, a distance of90 mi due north.Due to bad weather,the plane is redirected attake-offto a heading N 60? W(60? west ofnorth). Afterflying57 mi,the plane is directed to turn northeast and fly directly toward City B.To the nearest tenth,how many miles did the plane fly on the last leg of the trip?

For each triangle shown below,determine whether you would use the Law

ofSines or Law ofCosinesto find the value ofx.Then find the value ofx to the nearest tenth.

19.

City B

90 mi

23. A 15-ft water slide has a 9.5-ft ladder which meets the slide

at a 95? angle.To the nearest tenth, what is the distance

between the end ofthe slide and the bottom ofthe ladder?

24. Flags The dimensions ofa triangular flag are 18ft by 25ft by 27ft. To the nearest tenth, what is the measure ofthe angle formed by the two shorter sides?

Challenge 25. Parallelogram QRSThas a perimeter of62 mm.To the nearest tenth, what is the length of TB?

0

26. Surveying A surveyor measures the distance to the base ofa monument to be 12.4 meters at an angle ofelevation of 11?.At an angle ofelevation of26?,the distance to the top ofthe monumentis 13.3 meters.Whatis the height ofthe

monument to the nearest tenth?

13.3 m 12.4m

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Lesson 8-6 Law of Cosines

531

27. An isosceles triangleXVZhas abase of12in. and a heightof8in.To the nearest tenth, what are the measures ofthe angles?

28. Open-Ended Describe a situation in which you are given three measures ofa triangle but are unable to solve the triangle for the other three measures.

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Apply What You've Learned

Look back at the information on page 489 about the fire in a state forest. The diagram is shown again below.

MATHEMATICAL

PRACTICES

MR 1,MP2

W

E

Lookout Tower A

2000 m

S

Lookout Tower B

For parts(a)and(b),refer to the distances you found in the Apply WhatYou

Learned section in Lesson 8-5.

a. Use the Law of Cosines to verify the distance between Lookout Tower A

and the fire.

b. Use the Law of Cosines to verify the distance between Lookout Tower B

and the fire.

c. Write an equation that can be used to find how far west of Lookout

Tower B the fire is located.

532 Chapter 8 Right Triangles and Trigonometry

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