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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
8-5 Law of Sines and Law of Cosines
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Geometry
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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°
tan 103°  –
4.33
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B. cos 165°
cos 165°  –
0.97
C. sin 93°
sin 93°  1.00
8-5 Law of Sines and Law of Cosines
8.5 Practice (On a Separate Sheet of Paper)
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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).
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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°
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Cross Products Property
Divide both sides by sin 39.
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.
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8-5 Law of Sines and Law of Cosines
8.5 Practice (Continued)
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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.
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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
Law of Cosines
Substitute the
2
2
= 35 + 30 – 2(35)(30)cos 110°
given values.
XZ2  2843.2423
XZ  53.3
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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
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
8.5 Practice (Continued)
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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°
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0.31
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
477.2
a.
21.6
5. a = 16, b = 10, and mC = 110°. Find c.
6. a = 20, b = 15, and c = 8.3. Find mA.
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115°
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°
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