The Law of Sines and Law of Cosines CHAPTER 8.6. Concept.

The Law of Sines and Law of Cosines

CHAPTER 8.6

Concept

Example 1Law of Sines (AAS or ASA)

Find p. Round to the nearest tenth.

We are given measures of two angles and a nonincluded side, so use the Law of Sines to write a proportion.

Example 1

A. 4.6

B. 29.9

C. 7.8

D. 8.5

Find c to the nearest tenth.

Example 2Law of Sines (ASA)

Find x. Round to the nearest tenth.

6

x 57°

Example 2

A. 8

B. 10

C. 12

D. 14

Find x. Round to the nearest degree.

43°

x

Concept

Example 3Law of Cosines (SAS)

Find x. Round to the nearest tenth.

Use the Law of Cosines since the measures of two sides and the included angle are known.

Example 3

A. 25.1

B. 44.5

C. 22.7

D. 21.1

Find r if s = 15, t = 32, and mR = 40. Round to the nearest tenth.

Example 4Law of Cosines (SSS)

Find mL. Round to the nearest degree.

Law of Cosines

Simplify.

Example 4

A. 44°

B. 51°

C. 56°

D. 69°

Find mP. Round to the nearest degree.

Example 5 Indirect Measurement

AIRCRAFT From the diagram of the plane shown, determine the approximate width of each wing. Round to the nearest tenth meter.

Example 5

A. 93.5 in.

B. 103.5 in.

C. 96.7 in.

D. 88.8 in.

The rear side window of a station wagon has the shape shown in the figure. Find the perimeter of the window if the length of DB is 31 inches. Round to the nearest tenth.

Example 6Solve a Triangle

Solve triangle PQR. Round to the nearest degree.

Since the measures of three sides are given (SSS), use the Law of Cosines to find mP.

p2 = r2 + q2 – 2pq cos P Law of Cosines

82 = 92 + 72 – 2(9)(7) cos P p = 8, r = 9, and q = 7

Example 6

A. mR = 82, mS = 58, mT = 40

B. mR = 58, mS = 82, mT = 40

C. mR = 82, mS = 40, mT = 58

D. mR = 40, mS = 58, mT = 82

Solve ΔRST. Round to the nearest degree.

Concept