8-5
8-5 Law
LawofofSines
Sinesand
andLaw
LawofofCosines
Cosines
Warm Up
Lesson Presentation
Lesson Quiz
Holt
Geometry
Holt
Geometry
8-5 Law of Sines and Law of Cosines
Warm Up
1. What is the third angle measure in a triangle with
angles measuring 65° and 43°?
72°
Find each value. Round trigonometric
ratios to the nearest hundredth and angle
measures to the nearest degree.
2. sin 73° 0.96
3. cos 18° 0.95 4. tan 82° 7.12
5. sin-1 (0.34)
6. cos-1 (0.63)
20°
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51°
7. tan-1 (2.75)
70°
8-5 Law of Sines and Law of Cosines
Objective
Use the Law of Sines and the Law of
Cosines to solve triangles.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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°
tan 103°  –4.33 cos 165°  –0.97
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C. sin 93°
sin 93°  1.00
8-5 Law of Sines and Law of Cosines
Check It Out! Example 1
Use a calculator to find each trigonometric
ratio. Round to the nearest hundredth.
a. tan 175°
tan 175°  –0.09
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b. cos 92°
cos 92°  –0.03
c. sin 160°
sin 160°  0.34
8-5 Law of Sines and Law of Cosines
You can use the altitude of a triangle to find a
relationship between the triangle’s side lengths.
In ∆ABC, let h represent the length
of the altitude from C to
From the diagram,
,
and
By solving for h, you find that h = b sin A and h = a
sin B. So b sin A = a sin B, and
.
You can use another altitude to show that these
ratios equal
Holt Geometry
8-5 Law of Sines and Law of Cosines
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).
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
FG sin 39° = 40 sin 32°
Cross Products Property
Divide both sides by sin 39.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
NP sin 39° = 22 sin 88° Cross Products Property
Divide both sides by sin 39°.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
10 sin L = 6 sin 125°
Cross Products Property
Use the inverse sine
function to find mL.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
7.6 sin X = 4.3 sin 50°
Cross Products Property
Use the inverse sine
function to find mX.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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°
mA + 67° + 44° = 180°
mA = 69°
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Prop of ∆.
Substitute the given values.
Simplify.
8-5 Law of Sines and Law of Cosines
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.
AC sin 69° = 18 sin 67°
Cross Products Property
Divide both sides by sin 69°.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and 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).
Holt Geometry
8-5 Law of Sines and Law of Cosines
Helpful Hint
The angle referenced in the Law of Cosines is
across the equal sign from its corresponding
side.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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
= 352 + 302 – 2(35)(30)cos 110°
XZ2  2843.2423
XZ  53.3
Holt Geometry
Law of Cosines
Substitute the
given values.
Simplify.
Find the square
root of both
sides.
8-5 Law of Sines and Law of Cosines
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
72 = 132 + 112 – 2(13)(11)cos T
49 = 290 – 286 cosT
–241 = –286 cosT
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Law of Cosines
Substitute the
given values.
Simplify.
Subtract 290
both sides.
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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
= 182 + 162 – 2(18)(16)cos 21°
DE2  42.2577
DE  6.5
Holt Geometry
Law of Cosines
Substitute the
given values.
Simplify.
Find the square
root of both
sides.
8-5 Law of Sines and Law of Cosines
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
82 = 152 + 102 – 2(15)(10)cos K
64 = 325 – 300 cosK
–261 = –300 cosK
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Law of Cosines
Substitute the
given values.
Simplify.
Subtract 325
both sides.
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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
= 102 + 42 – 2(10)(4)cos 34°
YZ2  49.6770
YZ  7.0
Holt Geometry
Law of Cosines
Substitute the
given values.
Simplify.
Find the square
root of both
sides.
8-5 Law of Sines and Law of Cosines
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
2
2
2
9.6 = 5.9 + 10.5 – 2(5.9)(10.5)cos R
given values.
92.16 = 145.06 – 123.9cosR
–52.9 = –123.9 cosR
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Simplify.
Subtract 145.06
both sides.
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
Helpful Hint
Do not round your answer until the final step of
the computation. If a problem has multiple steps,
store the calculated answers to each part in your
calculator.
Holt Geometry
8-5 Law of Sines and Law of Cosines
Example 4: Sailing Application
A sailing club has planned a
triangular racecourse, as shown in
the diagram. How long is the leg of
the race along BC? How many
degrees must competitors turn at
point C? Round the length to the
nearest tenth and the angle
measure to the nearest degree.
Holt Geometry
8-5 Law of Sines and Law of Cosines
Example 4 Continued
Step 1 Find BC.
BC2 = AB2 + AC2 – 2(AB)(AC)cos A
=
3.92
+
BC2  7.7222
BC  2.8 mi
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3.12
Law of Cosines
Substitute the
– 2(3.9)(3.1)cos 45°
given values.
Simplify.
Find the square
root of both
sides.
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
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.
31 m
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8-5 Law of Sines and Law of Cosines
Check It Out! Example 4 Continued
Step 1 Find the length of the cable.
AC2 = AB2 + BC2 – 2(AB)(BC)cos B
=
312
+
562
– 2(31)(56)cos 100°
AC2  4699.9065
AC 68.6 m
Holt Geometry
Law of Cosines
Substitute the
given values.
Simplify.
Find the square
root of both
sides.
8-5 Law of Sines and Law of Cosines
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.
Holt Geometry
8-5 Law of Sines and Law of Cosines
Lesson Quiz: Part I
Use a calculator to find each trigonometric
ratio. Round to the nearest hundredth.
1. tan 154°
–0.49
2. cos 124°
–0.56
3. sin 162°
0.31
Holt Geometry
8-5 Law of Sines and Law of Cosines
Lesson Quiz: Part II
Use ΔABC for Items 4–6. 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. 477.2
5. a = 16, b = 10, and mC = 110°. Find c. 21.6
6. a = 20, b = 15, and c = 8.3. Find mA. 115°
Holt Geometry
8-5 Law of Sines and Law of Cosines
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°
Holt Geometry