Law of Sines & Law of Cosines Sections 6-1/2. 2 An oblique triangle is a triangle that has no right...
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Transcript of Law of Sines & Law of Cosines Sections 6-1/2. 2 An oblique triangle is a triangle that has no right...
Law of Sines &Law of Cosines
Sections 6-1/2
2
An oblique triangle is a triangle that has no right angles.
To solve an oblique triangle, you need to know the measure of at least one side and the measures of any other two parts of the triangle – two sides, two angles, or one angle and one side.
C
BA
ab
c
3
The following cases are considered when solving oblique triangles.
1. Two angles and any side (AAS or ASA)
2. Two sides and an angle opposite one of them (SSA)
3. Three sides (SSS)
4. Two sides and their included angle (SAS)
A
C
c
A
B
c
a
cb
C
c
a
c
aB
4
The first two cases can be solved using the Law of Sines. (The last two cases can be solved using the Law of Cosines.)
Law of Sines
If ABC is an oblique triangle with sides a, b, and c, then
.sin sin sin
a b cA B C
Acute Triangle
C
BA
bh
c
a
C
BA
bh
c
a
Obtuse Triangle
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5
Find the remaining angle and sides of the triangle.Example (ASA):
sin sina b
A B
The third angle in the triangle is A = 180 – A – B = 180 – 10 – 60 = 110.
C
BA
b
c
60
10
a = 4.5 ft
110
4.5110 60sin sin
b
Use the Law of Sines to find side b and c.
4.15 feetb
4.15 ft
sin sina c
A C
4.5110 10sin sin
c
0.83 feetc
0.83 ft
6
Use the Law of Sines to solve the triangle.A = 110, a = 125 inches, b = 100 inches
Example (SSA):
sin sina b
A B
125 100110sin sin B
48.74B
C 180 – 110 – 48.74 sin sin
a cA C
125110 21.26sin sin
c 48.23 inchesc
C
BA
b = 100 in
c
a = 125 in
110 48.74
21.26
48.23 in
= 21.26
7
Use the Law of Sines to solve the triangle.A = 76, a = 18 inches, b = 20 inches
Example (SSA):
sin sina b
A B
18 2076sin sin B
sin 1.078B
There is no angle whose sine is 1.078.
There is no triangle satisfying the given conditions.
C
AB
b = 20 ina = 18 in
76
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8
Use the Law of Sines to solve the triangle.A = 58, a = 11.4 cm, b = 12.8 cm
Example (SSA):
sin sina b
A B
11.4 12.858sin sin B
72.2
10.3 cm
Two different triangles can be formed.
49.8
a = 11.4 cm
C
AB1
b = 12.8 cm
c
58
Example continues.
1 72.2B
12.si
8n si49.8 2.2n 7
c
10.3c C 180 – 58 – 72.2 = 49.8
sin sinc b
C B
Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9
Use the Law of Sines to solve the second triangle.A = 58, a = 11.4 cm, b = 12.8 cm
Example (SSA) continued:
B2 180 – 72.2 = 107.8
107.8
C
AB2
b = 12.8 cm
c
a = 11.4 cm
58
14.2
3.3 cm
72.2
10.3 cm
49.8
a = 11.4 cm
C
AB1
b = 12.8 cm
c
58
C 180 – 58 – 107.8 = 14.2
12.si
8n si14.2 2.2n 7
c
3.3c
sin sinc b
C B
10
Area of an Oblique Triangle
C
BA
b
c
aFind the area of the triangle.A = 74, b = 103 inches, c = 58 inches
Example:
74
103 in
58 in 1Area = sin
2bc A
1 1 1Area sin sin sin2 2 2
bc A ab C ac B
103 51= ( )( )8 sin2
74
2871.29 sq. inches
11
The following cases are considered when solving oblique triangles.
1. Two angles and any side (AAS or ASA)
2. Two sides and an angle opposite one of them (SSA)
3. Three sides (SSS)
4. Two sides and their included angle (SAS)
A
C
c
A
B
c
a
cb
C
c
a
c
aB
12
The last two cases (SSS and SAS) can be solved using the Law of Cosines. (The first two cases can be solved using the Law of Sines.)
Law of CosinesStandard Form
2 2 2 2 cosa b c bc A
Alternative Form2 2 2
cos2
b c aAbc
2 2 2 2 cosb a c ac B 2 2 2
cos2
a c bBac
2 2 2 2 cosc a b ab C 2 2 2
cos2
a b cCab
13
Find the three angles of the triangle.Example:
C
BA
86
12
2 2 2cos
2a b cC
ab
Find the angle opposite the longest side first.
2 2 2122(
8 68 6)( )
64 36 14496
4496
117.3C
117.12
sin 3 in6
s BLaw of Sines: 26.4B
180 117.3 26.4A
36.3
117.3
26.4
36.3
14
Solve the triangle.Example:
67.8
759.9 6.2
sin sin A Law of Sines: 37.2A
180 75 37.2 67.8C
37.2
C
BA
6.2
759.52 2 2 2 cosb a c ac B
2 2( ) ( ) 2( )6.2 9.5 6.2 9.5( 7s 5)co
38.44 90.25 (117.8)(0.25882)
98.209.9b
9.9
Law of Cosines:
15
Homework
• WS 13-1