Five-Minute Check (over Lesson 8–6) Mathematical Practices 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

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 5-Minute Check 1

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 5-Minute Check 2

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. 5000 m B. 5750 m C. 6250 m D. 6500 m 5-Minute Check 3

Find the area of the triangle shown. A. 72.4 sq ft B. 65 sq ft C. 62.5 sq ft D. 56.4 sq ft 5-Minute Check 4

Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. MP

You solved triangles by using the Law of Sines. Use the Law of Cosines to solve triangles. Choose methods to solve triangles. Then/Now

Law of Cosines Vocabulary

Concept

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 = 72 + 102 – 2(7)(10) cos 73° a = 7, b = 10, and C = 73º c2  108.1 Simplify using a calculator. c  10.4 Take the square root of each side. Example 1

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

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

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 Example 1

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 = 92 + 72 – 2(9)(7) cos C a = 9, b = 7, and c = 12 Example 2

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 72 from each side. Divide each side by –126. – 0.1111  cos C Use a calculator. 96.4°  mC Use the cos–1 function. Example 2

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

Step 3 Find the measure of angle A. Solve a Triangle Given Three Sides Step 3 Find the measure of angle A. mA = 180° – (35.4° + 96.4°) or 48.2° Answer: Therefore, mA  48.2°, mB  35.4°, mC  96.4°. Example 2

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° Example 2

Concept

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

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

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 = 752 + 1102 – 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

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 mB ≈ 41° and mC ≈ 73°. Since mB < mA < mC and b < a < c, the solution is reasonable. Example 3

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 Example 3