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8-5

Law of Sines and Law of Cosines

Warm Up

Lesson Presentation

Lesson Quiz

Holt Geometry

Warm Up 1. What is the third angle measure in a

triangle with angles measuring 65 and

43? Find each value. Round trigonometric ratios

to the nearest hundredth and angle measures to

the nearest degree. 2. sin 73 3. cos 18 4.

tan 82 5. sin-1 (0.34) 6. cos-1 (0.63) 7.

tan-1 (2.75)

72

0.96

0.95

7.12

20

51

70

Objective

Use the Law of Sines and the Law of Cosines to

solve triangles.

In this lesson, you will learn to solve any

triangle. To do so, you will need to calculate

trigonometric ratios for angle measures up to

180. You can use a calculator to find these

values.

Example 1 Finding Trigonometric Ratios for

Obtuse Angles

Use your calculator to find each trigonometric

ratio. Round to the nearest hundredth.

A. tan 103

B. cos 165

C. sin 93

tan 103 ? 4.33

cos 165 ? 0.97

sin 93 ? 1.00

Check It Out! Example 1

Use a calculator to find each trigonometric

ratio. Round to the nearest hundredth.

a. tan 175

b. cos 92

c. sin 160

tan 175 ? 0.09

cos 92 ? 0.03

sin 160 ? 0.34

You can use the altitude of a triangle to find a

relationship between the triangles side lengths.

You can use the Law of Sines to solve a triangle

if you are given two angle measures and any

side length (ASA or AAS) or two side lengths

and a non-included angle measure (SSA).

Example 2A Using the Law of Sines

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

FG

Law of Sines

Substitute the given values.

Cross Products Property

FG sin 39 40 sin 32

Divide both sides by sin 39?.

Example 2B Using the Law of Sines

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?Q

Law of Sines

Substitute the given values.

Multiply both sides by 6.

Use the inverse sine function to find m?Q.

Check It Out! Example 2a

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

NP

Law of Sines

Substitute the given values.

Cross Products Property

NP sin 39 22 sin 88

Divide both sides by sin 39.

Check It Out! Example 2b

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?L

Law of Sines

Substitute the given values.

Cross Products Property

10 sin L 6 sin 125

Use the inverse sine function to find m?L.

Check It Out! Example 2c

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?X

Law of Sines

Substitute the given values.

Cross Products Property

7.6 sin X 4.3 sin 50

Use the inverse sine function to find m?X.

Check It Out! Example 2d

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

AC

m?A m?B m?C 180

Prop of ?.

Substitute the given values.

m?A 67 44 180

m?A 69

Simplify.

Check It Out! Example 2D Continued

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

Law of Sines

Substitute the given values.

Cross Products Property

AC sin 69 18 sin 67

Divide both sides by sin 69.

The Law of Sines cannot be used to solve every

triangle. If you know two side lengths and the

included angle measure or if you know all three

side lengths, you cannot use the Law of Sines.

Instead, you can apply the Law of Cosines.

You can use the Law of Cosines to solve a

triangle if you are given two side lengths and

the included angle measure (SAS) or three side

lengths (SSS).

(No Transcript)

Example 3A Using the Law of Cosines

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

XZ

XZ2 XY2 YZ2 2(XY)(YZ)cos Y

Law of Cosines

Substitute the given values.

352 302 2(35)(30)cos 110

XZ2 ? 2843.2423

Simplify.

Find the square root of both sides.

XZ ? 53.3

Example 3B Using the Law of Cosines

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?T

RS2 RT2 ST2 2(RT)(ST)cos T

Law of Cosines

Substitute the given values.

72 132 112 2(13)(11)cos T

49 290 286 cosT

Simplify.

Subtract 290 both sides.

241 286 cosT

Example 3B Continued

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?T

241 286 cosT

Solve for cosT.

Use the inverse cosine function to find m?T.

Check It Out! Example 3a

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

DE

DE2 EF2 DF2 2(EF)(DF)cos F

Law of Cosines

Substitute the given values.

182 162 2(18)(16)cos 21

DE2 ? 42.2577

Simplify.

Find the square root of both sides.

DE ? 6.5

Check It Out! Example 3b

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?K

JL2 LK2 KJ2 2(LK)(KJ)cos K

Law of Cosines

Substitute the given values.

82 152 102 2(15)(10)cos K

64 325 300 cosK

Simplify.

Subtract 325 both sides.

261 300 cosK

Check It Out! Example 3b Continued

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?K

261 300 cosK

Solve for cosK.

Use the inverse cosine function to find m?K.

Check It Out! Example 3c

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

YZ

YZ2 XY2 XZ2 2(XY)(XZ)cos X

Law of Cosines

Substitute the given values.

102 42 2(10)(4)cos 34

YZ2 ? 49.6770

Simplify.

Find the square root of both sides.

YZ ? 7.0

Check It Out! Example 3d

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?R

PQ2 PR2 RQ2 2(PR)(RQ)cos R

Law of Cosines

Substitute the given values.

9.62 5.92 10.52 2(5.9)(10.5)cos R

92.16 145.06 123.9cosR

Simplify.

Subtract 145.06 both sides.

52.9 123.9 cosR

Check It Out! Example 3d Continued

Find the measure. Round lengths to the nearest

tenth and angle measures to the nearest degree.

m?R

52.9 123.9 cosR

Solve for cosR.

Use the inverse cosine function to find m?R.

(No Transcript)

Example 4 Sailing Application

Example 4 Continued

Step 1 Find BC.

BC2 AB2 AC2 2(AB)(AC)cos A

Law of Cosines

Substitute the given values.

3.92 3.12 2(3.9)(3.1)cos 45

Simplify.

BC2 ? 7.7222

Find the square root of both sides.

BC ? 2.8 mi

Example 4 Continued

Step 2 Find the measure of the angle through

which competitors must turn. This is m?C.

Law of Sines

Substitute the given values.

Multiply both sides by 3.9.

Use the inverse sine function to find m?C.

Check It Out! Example 4

What if? Another engineer suggested using a

cable attached from the top of the tower to a

point 31 m from the base. How long would this

cable be, and what angle would it make with the

ground? Round the length to the nearest tenth and

the angle measure to the nearest degree.

Check It Out! Example 4 Continued

Step 1 Find the length of the cable.

AC2 AB2 BC2 2(AB)(BC)cos B

Law of Cosines

Substitute the given values.

312 562 2(31)(56)cos 100

Simplify.

AC2 ? 4699.9065

Find the square root of both sides.

AC ?68.6 m

Check It Out! Example 4 Continued

Step 2 Find the measure of the angle the cable

would make with the ground.

Law of Sines

Substitute the given values.

Multiply both sides by 56.

Use the inverse sine function to find m?A.

Lesson Quiz Part I

Use a calculator to find each trigonometric

ratio. Round to the nearest hundredth. 1. tan

154 2. cos 124 3. sin 162

0.49

0.56

0.31

Lesson Quiz Part II

Use ?ABC for Items 46. Round lengths to the

nearest tenth and angle measures to the nearest

degree. 4. m?B 20, m?C 31 and b

210. Find a. 5. a 16, b 10, and m?C 110.

Find c. 6. a 20, b 15, and c 8.3. Find

m?A.

477.2

21.6

115

Lesson Quiz Part III

7. An observer in tower A sees a fire 1554 ft

away at an angle of depression of 28. To the

nearest foot, how far is the fire from an

observer in tower B? To the nearest degree, what

is the angle of depression to the fire from tower

B?

1212 ft 37

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