Download 8-6 The Law of Sines and Law of Cosines 2-23

2/23/2016
Five-Minute Check (over Lesson 8–5)
CCSS
Then/Now
New Vocabulary
Theorem 8.10: Law of Sines
Example 1: Law of Sines (AAS or ASA)
Example 2: Law of Sines (ASA)
Theorem 8.11: Law of Cosines
Example 3: Law of Cosines (SAS)
Example 4: Law of Cosines (SSS)
Example 5: Real-World Example: Indirect Measurement
Example 6: Solve a Triangle
Concept Summary: Solving a Triangle
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Over Lesson 8–5
Name the angle of
depression in the figure.
Find the angle of elevation of the Sun when a
6-meter flagpole casts a 17-meter shadow.
After flying at an altitude of 575 meters, a
helicopter starts to descend when its ground
distance from the landing pad is 13.5 kilometers.
What is the angle of depression for this part of the
flight?
The top of a signal tower is 250 feet above sea
level. The angle of depression from the top of the
tower to a passing ship is 19°. How far is the foot
of the tower from the ship?
Over Lesson 8–5
Name the angle of
depression in the figure.
A. ∠URT
B. ∠SRT
C. ∠RST
D. ∠SRU
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Over Lesson 8–5
Find the angle of elevation of the Sun when a
6-meter flagpole casts a 17-meter shadow.
A. about 70.6°
B. about 60.4°
C. about 29.6°
D. about 19.4°
Over Lesson 8–5
After flying at an altitude of 575 meters, a
helicopter starts to descend when its ground
distance from the landing pad is 13.5 kilometers.
What is the angle of depression for this part of the
flight?
A. about 1.8°
B. about 2.4°
C. about 82.4°
D. about 88.6°
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Over Lesson 8–5
The top of a signal tower is 250 feet above sea
level. The angle of depression from the top of the
tower to a passing ship is 19°. How far is the foot
of the tower from the ship?
A. about 81.4 ft
B. about 236.4 ft
C. about 726 ft
D. about 804 ft
Over Lesson 8–5
Jay is standing 50 feet away from the Eiffel Tower
and measures the angle of elevation to the top of
the tower as 87.3°. Approximately how tall is the
Eiffel Tower?
A. 50 ft
B. 104 ft
C. 1060 ft
D. 4365 ft
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Content Standards
G.SRT.9 Derive the formula A = ab sin (C) for
the area of a triangle by drawing an auxiliary line
from a vertex perpendicular to the opposite side.
G.SRT.10 Prove the Laws of Sines and Cosines
and use them to solve problems.
Mathematical Practices
4 Model with mathematics.
1 Make sense of problems and persevere in
solving them.
You used trigonometric ratios to solve right
triangles.
• Use the Law of Sines to solve triangles.
• Use the Law of Cosines to solve triangles.
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• Law of Sines
• Law of Cosines
Alternatively: = = 6
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Law of Sines (AAS or ASA)
Find p. Round to the nearest tenth.
=
(8)17
=
29
=
≈ 4.8
Answer: p ≈ 4.8
Find c to the nearest tenth.
A. 4.6
B. 29.9
C. 7.8
D. 8.5
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Law of Sines (ASA)
Find x. Round to the nearest tenth.
Notice: ∠ = 73°
"
=
! !
=
(6)50
# =
73
6
57°
73°
x
# ≈ 4.8
Answer: x ≈ 4.8
Find x. Round to the nearest degree.
A. 8
B. 10
x
C. 12
D. 14
43°
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Notice: Since cos 90°= 0 this is a more generalized
version of the Pythagorean theorem!
Law of Cosines (SAS)
Find x. Round to the nearest tenth.
# ' = ( ' + * ' − 2(*",-.
# ' = (11)' +(25)' − 2(11)(25)",-45
#=
11
'
+ 25
'
− 2 11 25 cos 45
# ≈18.9
Answer: x ≈ 18.9
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Find r if s = 15, t = 32, and m∠
∠R = 40. Round to the
nearest tenth.
A. 25.1
B. 44.5
C. 22.7
D. 21.1
Law of Cosines (SSS)
Find m∠
∠L. Round to the nearest degree.
3' = '
+ '
− 2 cos 4
24 ' = 27 ' + 5 ' − 2 27 5 cos 4
2 27 5 ",-4 = 27 ' + 5 ' − 24 '
(27)' +(5)' − (24)'
",-4 =
2 27 5
27 ' + 5 ' − 24 '
≈ 49°
4 = ",- 56
2 27 5
Answer: ∠4 ≈ 49°
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Find m∠
∠P. Round to the nearest degree.
A. 44°
B. 51°
C. 56°
D. 69°
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