Section 1.6
Law of Cosines
Objectives:
1. To prove the law of cosines.
2. To solve triangles using the law
of cosines.
Law of Cosines
For any triangle ABC, where side
lengths opposite angles A, B, and C
are a, b, and c respectively, then
a2 = b2 + c2 – 2bc cos A.
Alternate forms of the Law of
Cosines
a2 = b2 + c2 – 2bc cos A
b2 = a2 + c2 – 2ac cos B
c2 = a2 + b2 – 2ab cos C
To apply the Law of Cosines,
you must know either the
measures of all three sides or
the measures of two sides and
the included angle.
EXAMPLE 1 Solve FDG if f = 8, d
= 4, and g = 10.
f2 = d2 + g2 – 2dg cos F
82 = 42 + 102 – 2(4)(10) cos F
64 = 16 + 100 – 80 cos F
64 = 116 – 80 cos F
-52 = -80 cos F
mF = cos-1 (0.65) ≈ 49.46 (4928΄)
EXAMPLE 1 Solve FDG if f = 8, d
= 4, and g = 10.
g2 = d2 + f2 – 2df cos G
102 = 42 + 82 – 2(4)(8) cos G
100 = 16 + 64 – 64 cos G
100 = 80 – 64 cos G
20 = -64 cos G
mG = cos-1 (-0.3125) ≈ 108.21
(10813΄)
EXAMPLE 1 Solve FDG if f = 8, d
= 4, and g = 10.
mD = 180 – (mF + mG)
mD = 180 – (49.46 + 108.21)
mD = 180 – 157.67
mD ≈ 22.33 (2219΄)
EXAMPLE 1 Solve FDG if f = 8, d
= 4, and g = 10.
mF = 4928΄
mD = 2219΄
mG = 10813΄
f=8
d=4
g = 10
EXAMPLE 2 Solve ABC if A =
63, b = 12, and c = 9.
a2 = b2 + c2 – 2bc cos A
a2 = 122 + 92 – 2(12)(9) cos 63
a2 = 144 + 81 – 216(.4540)
a2 = 225 – 98.06
a ≈ 11.3
EXAMPLE 2 Solve ABC if A =
63, b = 12, and c = 9.
Use the law of sines to solve for C
since C must be the smallest angle.
EXAMPLE 2 Solve ABC if A =
63, b = 12, and c = 9.
a
c
=
sin A
sin C
11.3
9
=
sin 63 sin C
9(sin 63)
sin C =
11.3
mC = sin-1 (0.70965) ≈ 45.2
EXAMPLE 2 Solve ABC if A =
63, b = 12, and c = 9.
mB = 180 – (mA + mC)
mB = 180 – (63 + 45.2)
mB = 180 – 108.2
mB ≈ 71.8
EXAMPLE 2 Solve ABC if A =
63, b = 12, and c = 9.
mA = 63
mB = 71.8
mC = 45.2
a = 11.3
b = 12
c=9
Practice: Find the mA if a = 38, b
= 48, and c = 68. Round to the nearest
degree.
a2 = b2 + c2 – 2bc cos A
382 = 482 + 682 – 2(48)(68) cos A
1444 = 2304 + 4624 – 6528 cos A
-5484 = -6528 cos A
mA = cos-1 (0.8401) ≈ 33
Practice: Find the mC if a = 38, b
= 48, and c = 68. Round to the nearest
degree.
c2 = a2 + b2 – 2ab cos C
682 = 382 + 482 – 2(38)(48) cos C
4624 = 1444 + 2304 – 3648 cos C
876 = -3648 cos C
mC = cos-1 (-0.2401) ≈ 104
Homework
pp. 32-34
►A. Exercises
1. a = 6, b = 5, c = 8
62 = 52 + 82 – 2(5)(8)(cos A)
36 = 25 + 64 – 80 cos A
36 = 89 – 80 cos A
-53 = -80 cos A
cos A = 0.6625
mA = cos-1 (0.6625)
mA ≈ 48.5°
►A. Exercises
1. a = 6, b = 5, c = 8
52 = 62 + 82 – 2(6)(8)(cos B)
25 = 36 + 64 – 96 cos B
25 = 100 – 96 cos B
-75 = -96 cos B
cos B = 0.78125
mB = cos-1 (0.78125)
mB ≈ 38.6°
►A. Exercises
1. a = 6, b = 5, c = 8
mC = 180° - mA - mB
mC = 180° - 48.5° - 38.6°
mC ≈ 92.9°
►A. Exercises
1.
a =6
b =5
c =8
mA ≈ 48.5
mB ≈ 38.6
mC ≈ 92.9
►A. Exercises
3. b = 26, c = 18, mA = 64°
a2 = 262 + 182 – 2(26)(18)(cos 64°)
a2 = 676 + 324 – 936(0.4384)
a2 = 1000 – 410.3
a2 = 589.7
a ≈ 24.3
►A. Exercises
3. b = 26, c = 18, mA = 64°
262 = 24.32 + 182 – 2(24.3)(18)(cos B)
676 = 589.7 + 324 – 874.2(cos B)
676 = 913.7 – 874.2(cos B)
-237.7 = -874.2(cos B)
cos B = 0.2719
mB = cos-1 (0.2719)
mB ≈ 74.2°
►A. Exercises
3. b = 26, c = 18, mA = 64°
mC = 180° - mA - mB
mC = 180° - 64° - 74.2°
mC ≈ 41.8°
►A. Exercises
3.
a ≈ 24.3
b = 26
c = 18
mA = 64
mB ≈ 74.2
mC ≈ 41.8
►A. Exercises
7. A = 19.5°, B = 92°, c = 28
1. Basic trig ratios
2. Law of sines
3. Law of cosines
►A. Exercises
9. A = 60°, B = 90°, b = 10
1. Basic trig ratios
2. Law of sines
3. Law of cosines
►B. Exercises
11. A radio antenna is placed on the top
of a 200-foot office building. The
angle of elevation from a parking lot
to the top of the antenna is 21°. The
angle of depression looking from the
bottom of the antenna to the lot is
10°. What is the height of the
antenna?
►B. Exercises
11.
10°
200 ft
21°
►B. Exercises
11.
69°
100°
200 ft
x
11°
►B. Exercises
11.
200
sin10 =
x
200
x=
sin10
x ≈ 1151.75
200 ft
x
10°
►B. Exercises
11.
69°
x
100°
1151.75
x
=
sin69
sin11
1151.75(sin11)
x=
sin69
x ≈ 235.4
200 ft
11°
■ Cumulative Review
21. Convert 5 radians to degrees.
■ Cumulative Review
22. Give the reference angle for -470°.
■ Cumulative Review
23. Write 3 reciprocal ratios.
■ Cumulative Review
24. In ∆ABC, find b if B = 27°, a = 8,
and A = 90°
■ Cumulative Review
25. In ∆ABC, find b if B = 27°, a = 8,
and A = 20°