1. Splash Screen

2. Five-Minute Check (over Lesson 13–4) Then/Now New Vocabulary
Key Concept: Law of Cosines
Example 1: Solve a Triangle Given Two Sides and the Included Angle
Example 2: Solve a Triangle Given Three Sides
Concept Summary: Solving Oblique Triangles
Example 3: Real-World Example: Use the Law of Cosines
Lesson Menu

3. Find the area of ΔABC if A = 68°, c = 12 feet, and b = 16 feet.
A. 89 ft2
B. 93 ft2
C. 96 ft2
D. 102 ft2 A B C D
5-Minute Check 1

4. Solve the triangle. Round measures of sides to the nearest tenth and measures of angles to the nearest degree.
A. A = 24°, C = 31°, c = 4
B. A = 31°, C = 24°, c = 4
C. A = 24°, C = 31°, c = 6
D. A = 31°, C = 24°, c = 6 A B C D
5-Minute Check 2

5. Two towns are viewed from a 5000-meter peak at angles of depression of 48° and 26°. What is the distance d between the towns? A B C D
A m
B m
C m
D m
5-Minute Check 3

6. A B C D Find the area of the triangle shown. A. 72.4 sq ft B. 65 sq ft
C sq ft
D sq ft A B C D
5-Minute Check 4

7. You solved triangles by using the Law of Sines. (Lesson 13–4)
Use the Law of Cosines to solve triangles.
Choose methods to solve triangles.
Then/Now

8. Law of Cosines
Vocabulary

9. Concept

10. Step 1 Use the Law of Cosines to find c.
Solve a Triangle Given Two Sides and the Included Angle
Step 1 Use the Law of Cosines to find c.
c2 = a2 + b2 – 2ab cos C Law of Cosines
c2 = – 2(7)(10) cos 73° a = 7, b = 10, and C = 73
c2  Simplify using a calculator.
c  10.4 Take the square root of each side.
Example 1

11. Step 2 Use the Law of Sines to find the measure of angle A.
Solve a Triangle Given Two Sides and the Included Angle
Step 2 Use the Law of Sines to find the measure of angle A.
Law of Sines
A = 7, C = 73°, and c  10.4
Multiply each side by 7.
Use a calculator.
Use the sin–1 function.
Example 1

12. Step 3 Find the measure of angle B. B = 180° – (40° + 73°) or 67°
Solve a Triangle Given Two Sides and the Included Angle
Step 3 Find the measure of angle B.
B = 180° – (40° + 73°) or 67°
Answer: Therefore, c  10.4, A  40°, and B  67°.
Example 1

13. A B C D Solve ΔABC. A. A  49°, B  67°, c  5.5
B. A  35°, B  81°, c  12.3
C. A  62°, B  54°, c  9.1
D. A  37°, B  79°, c  11.9 A B C D
Example 1

14. c2 = a2 + b2 – 2ab cos C Law of Cosines
Solve a Triangle Given Three Sides
Solve ΔABC.
Step 1 Use the Law of Cosines to find the measure of the largest angle first, angle C.
c2 = a2 + b2 – 2ab cos C Law of Cosines
122 = – 2(9)(7) cos C a = 9, b = 7, and c = 12
Example 2

15. 122 – 92 – 72 = –2(9)(7) cos C Subtract 92 and 72 from each side.
Solve a Triangle Given Three Sides
122 – 92 – 72 = –2(9)(7) cos C Subtract 92 and from each side.
Divide each side by –126.
–  cos C Use a calculator.
96°  C Use the cos– function.
Example 2

16. Step 2 Use the Law of Sines to find the measure of angle B.
Solve a Triangle Given Three Sides
Step 2 Use the Law of Sines to find the measure of angle B.
Law of Sines
b = 7, C  96°, and c = 12
Multiply each side by 7.
Use a calculator.
Use the sin–1 function.
Example 2

17. Step 3 Find the measure of angle A. A = 180° – (35° + 96°) or 48°
Solve a Triangle Given Three Sides
Step 3 Find the measure of angle A.
A = 180° – (35° + 96°) or 48°
Answer: Therefore, A  48°, B  35°, and C  96°.
Example 2

18. A B C D Solve ΔABC. A. A  11°, B  113°, C  56°
B. A  14°, B  42°, C  124°
C. A  13°, B  113°, C  54°
D. A  6°, B  59°, C  115° A B C D
Example 2

19. Concept

20. Use the Law of Cosines
AIRPORT Two pilots in a stationary airplane look 38° to the left of their runway and see a bus 75 feet away. They look 28° to the right of their runway and see a truck 110 feet away. How far apart are the bus and the truck?
Understand You know the angles formed when the pilots look left and when they look right. You also know how far away the bus and the truck are from the pilots.
Example 3

21. Use the Law of Cosines
Plan Use the information to draw and label a diagram. Since two sides and the included angle of a triangle are given, you can use the Law of Cosines to solve the problem.
Example 3

22. a2 = b2 + c2 – 2bc cos A Law of Cosines
Use the Law of Cosines
Solve
a2 = b2 + c2 – 2bc cos A Law of Cosines
a2 = – 2(75)(110) cos 66° b = 75, c = 110, and A = 66°
a2 = 11,014 Use a calculator.
a ≈ 105 Find the positive value of a.
Example 3

23. Answer: So, the bus and the truck are about 105 feet apart.
Use the Law of Cosines
Answer: So, the bus and the truck are about 105 feet apart.
Check Using the Law of Sines, you can find that B ≈ 41° and C ≈ 73°. Since B < A < C and b < a < c, the solution is reasonable.
Example 3

24. DRIVING A driver in a stationary car looks 42° to the left of her lane and sees a tree 70 feet away. She looks 32° to the right of the lane and sees a building 103 feet away. How far apart are the tree and the building?
A. about 107 ft
B. about 111 ft
C. about 116 ft
D. about 122 ft A B C D
Example 3

25. End of the Lesson