Transcript File

The Law of Sines and the Law of
Cosines
Lesson 8.5
Objective
Use the Law of Sines to solve triangles.
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Ex1: Finding Trigonometric Ratios for Obtuse Angles
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.
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
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 2: 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
FG sin 39° = 40 sin 32°
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Check It Out! Example 3
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 ∆.
Check It Out! Example 3 Continued
Find the measure. Round lengths to
the nearest tenth and angle
measures to the nearest degree.
69°
Law of Sines
AC sin 69° = 18 sin 67°
Divide both sides by sin
69°.
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Check It Out! Example 4
Find the measure. Round lengths to the nearest
tenth and angle measures to the nearest
degree.
mX
Law of Sines
7.6 sin X = 4.3 sin 50°
Use the inverse sine
function to find mX.
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Example 5: 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
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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
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Lesson Quiz
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
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8.5 Day Two
Law of Sines
Objective
Use the Law of Cosines to solve triangles.
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The Law of Sines cannot be used to solve every ∆.
Instead, you can apply the Law of Cosines.
You can use the Law of Cosines to solve a ∆ if you are given:
• two side lengths and the included angle measure (SAS)
or
• three side lengths (SSS).
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Example 1: 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
XZ2 = 352 + 302 – 2(35)(30)cos 110° Substitute
XZ2  2843.2423
XZ  53.3
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Example 2
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
= 182 + 162 – 2(18)(16)cos 21°
DE2  42.2577
DE  6.5
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Example 3: 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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Example 4
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.
Example 4 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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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.
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Example 5
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.
Step 1 Find the length of the cable.
AC2 = AB2 + BC2 – 2(AB)(BC)cos B
AC2= 312 + 562 – 2(31)(56)cos 100°
AC2  4699.9065
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AC 68.6 m
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31 m
Example 5 Continued
Step 2 Find the measure of the angle the cable
would make with the ground.
Law of Sines
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31 m
Lesson Quiz
Use ΔABC for Items 4–6. Round lengths to
the nearest tenth and angle measures to the
nearest degree.
1. a = 20, b = 15, and c = 8.3. Find mA.
115°
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Lesson Quiz
2. 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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