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Section 7.1
The Law of Sines
A
B
Ca
c
b
c
C
b
B
a
A sinsinsin
Answers on next slide.
c
C
b
B
a
A sinsinsin
Solve the triangle: 4,80,45 bCB
45
80A
ac
4Answers on next slide.
Section 7.3
The Law of Cosines
ab
cbaC
ac
bcaB
bc
acbA
2cos
2cos
2cos
222
222
222
A
B
Ca
c
b
8© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 1 Solving SAS Triangles
Solve triangle ABC with a = 15 inches, b = 10 inches, and C = 60º. Round each answer to the nearest tenth. Solution
Step 1 Find side c opposite the given angle C.
2 2 2
2 22
2
15 110 1
2 cos
2 c 6os
1225 100 2 15 10 175
2
175
0
1
0°
.2
5
3
c
c
c
ba ab C
c
9© 2011 Pearson Education, Inc. All rights reserved
Solution continued
1
sin sin
sinsin
10sin 60°sin
175
10sin 60°sin 40.9°
175
B C
b cb C
Bc
B
B
EXAMPLE 1 Solving the SAS Triangles
Step 2 Find the angle B opposite side b.
10© 2011 Pearson Education, Inc. All rights reserved
Solution continued
EXAMPLE 1 Solving the SAS Triangles
Step 3 Use the angle sum formula to find the third angle.
Step 4 The solution of triangle ABC is
180° 60° 40.9° 79.1°A
A ≈ 79.1° a = 15 inches
B ≈ 40.9° b = 10 inches
C = 60° c ≈ 13.2 inches
Set these up:1. B = 20o a = 10 c = 152. A = 25o b = 3 c = 63. C = 75o a = 5 b = 3
4. B = 45o a = 8 c = 825. A = 60o b = 3 c = 66. C = 30o a = 24 b =
123
Set these up
and solve:
12© 2011 Pearson Education, Inc. All rights reserved
EXAMPLE 3 Solving the SSS Triangles
Solve triangle ABC with a = 3.1 feet, b = 5.4 feet, and c = 7.2 feet. Round answers to the nearest tenth. Solution
Step 1 Because c is the longest side, find C. 2 2 2
2 2 2
2 cos
2 cos
c a b ab C
ab C a b c
2 2 22 2 2 5cos
7.40.39
2
3.1
3.12 5.4
.2cb
ab
aC
1cos 0.39 113°C
13© 2011 Pearson Education, Inc. All rights reserved
Solution continued
sin sinB C
b c
EXAMPLE 3 Solving the SSS Triangles
Step 2 Find B.
1
1
sinsin
sinsin
5.4sin113°sin 43.7°
7.2
b CB
cb C
Bc
14© 2011 Pearson Education, Inc. All rights reserved
Solution continued
Step 3 A ≈ 180º − 43.7º − 113º ≈ 23.3º
Step 4 Write the solution.
EXAMPLE 3 Solving the SSS Triangles
A ≈ 23.3° a = 3.1 feet
B ≈ 43.7° b = 5.4 feet
C ≈ 113° c = 7.2 feet
Example
a=5, b=7, c=8, find cosine of greatest angle.
15© 2011 Pearson Education, Inc. All rights reserved
