1. Section 1.6
Law of Cosines

2. Objectives:
1. To prove the law of cosines.
2. To solve triangles using the law
of cosines.

3. 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.

4. 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

5. 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.

6. EXAMPLE 1 Solve FDG if f = 8, d = 4, and g = 10.
f2 = d2 + g2 – 2dg cos F
82 = – 2(4)(10) cos F
64 = – 80 cos F
64 = 116 – 80 cos F
-52 = -80 cos F
mF = cos-1 (0.65) ≈ 49.46 (4928΄)

7. EXAMPLE 1 Solve FDG if f = 8, d = 4, and g = 10.
g2 = d2 + f2 – 2df cos G
102 = – 2(4)(8) cos G
100 = – 64 cos G
100 = 80 – 64 cos G
20 = -64 cos G
mG = cos-1 ( ) ≈  (10813΄)

8. EXAMPLE 1 Solve FDG if f = 8, d = 4, and g = 10.
mD = 180 – (mF + mG)
mD = 180 – (49.46 )
mD = 180 – 
mD ≈ 22.33 (2219΄)

9. EXAMPLE 1 Solve FDG if f = 8, d = 4, and g = 10.
mF = 4928΄ f = 8
mD = 2219΄ d = 4
mG = 10813΄ g = 10

10. EXAMPLE 2 Solve ABC if A = 63, b = 12, and c = 9.
a2 = b2 + c2 – 2bc cos A
a2 = – 2(12)(9) cos 63
a2 = – 216(.4540)
a2 = 225 – 98.06
a ≈ 11.3

11. 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.

12. EXAMPLE 2 Solve ABC if A = 63, b = 12, and c = 9.
sin A c
sin C =
11.3
sin 63 9
sin C =
9(sin 63)
11.3
sin C =
mC = sin-1 ( ) ≈ 45.2

13. EXAMPLE 2 Solve ABC if A = 63, b = 12, and c = 9.
mB = 180 – (mA + mC)
mB = 180 – (63 )
mB = 180 – 108.2
mB ≈ 71.8

14. EXAMPLE 2 Solve ABC if A = 63, b = 12, and c = 9.
mA = 63 a = 11.3
mB = 71.8 b = 12
mC = 45.2 c = 9

15. Practice: Find the mA if a = 38, b = 48, and c = 68
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 = – 2(48)(68) cos A
1444 = – 6528 cos A
-5484 = cos A
mA = cos-1 (0.8401) ≈ 33

16. Practice: Find the mC if a = 38, b = 48, and c = 68
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 = – 2(38)(48) cos C
4624 = – 3648 cos C
876 = cos C
mC = cos-1 ( ) ≈ 104

17. Homework pp

18. ►A. Exercises
1. a = 6, b = 5, c = 8
62 = – 2(5)(8)(cos A)
36 = – 80 cos A
36 = 89 – 80 cos A
-53 = -80 cos A
cos A =
mA = cos-1 (0.6625)
mA ≈ 48.5°

19. ►A. Exercises
1. a = 6, b = 5, c = 8
52 = – 2(6)(8)(cos B)
25 = – 96 cos B
25 = 100 – 96 cos B
-75 = -96 cos B
cos B =
mB = cos-1 ( )
mB ≈ 38.6°

20. ►A. Exercises
1. a = 6, b = 5, c = 8
mC = 180° - mA - mB
mC = 180° ° °
mC ≈ 92.9°

21. ►A. Exercises 1.
a = 6
b = 5
c = 8
mA ≈ 48.5
mB ≈ 38.6
mC ≈ 92.9

22. ►A. Exercises
3. b = 26, c = 18, mA = 64°
a2 = – 2(26)(18)(cos 64°)
a2 = – 936(0.4384)
a2 = 1000 – 410.3
a2 = 589.7
a ≈ 24.3

23. ►A. Exercises
3. b = 26, c = 18, mA = 64°
262 = – 2(24.3)(18)(cos B)
676 = – 874.2(cos B)
676 = – 874.2(cos B)
= (cos B)
cos B =
mB = cos-1 (0.2719)
mB ≈ 74.2°

24. ►A. Exercises
3. b = 26, c = 18, mA = 64°
mC = 180° - mA - mB
mC = 180° - 64° °
mC ≈ 41.8°

25. ►A. Exercises 3.
a ≈ 24.3
b = 26
c = 18
mA = 64
mB ≈ 74.2
mC ≈ 41.8

26. ►A. Exercises
7. A = 19.5°, B = 92°, c = 28
1. Basic trig ratios
2. Law of sines
3. Law of cosines

27. ►A. Exercises
9. A = 60°, B = 90°, b = 10
1. Basic trig ratios
2. Law of sines
3. Law of cosines

28. 11. A radio antenna is placed on the top
►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?

29. ►B. Exercises
11.
10°
200 ft
21°

30. ►B. Exercises
11.
69°
100° x
200 ft
11°

31. ►B. Exercises
sin10 =
200 x
11.
sin10
200
x =
x ≈ x
200 ft
10°

32. ►B. Exercises 11. x 69° 100° 1151.75 200 ft 11° sin69 1151.75 = sin11

33. ■ Cumulative Review
21. Convert 5 radians to degrees.

34. ■ Cumulative Review
22. Give the reference angle for -470°.

35. ■ Cumulative Review
23. Write 3 reciprocal ratios.

36. ■ Cumulative Review
24. In ∆ABC, find b if B = 27°, a = 8, and A = 90°

37. ■ Cumulative Review
25. In ∆ABC, find b if B = 27°, a = 8, and A = 20°