Law of Sines and Law of Cosines

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8-5
Law of Sines and Law of Cosines
Warm Up
Lesson Presentation
Lesson Quiz
Holt Geometry
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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
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Objective
Use the Law of Sines and the Law of Cosines to
solve triangles.
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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.
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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
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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
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You can use the altitude of a triangle to find a
relationship between the triangles side lengths.
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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).
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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?.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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).
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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
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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
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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.
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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
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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
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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.
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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
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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
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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.
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Example 4 Sailing Application
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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
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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.
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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.
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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
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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.
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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
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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
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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