1. Copyright © Cengage Learning. All rights reserved.
6.6
The Law of Cosines
Copyright © Cengage Learning. All rights reserved.

2. The Law of Cosines
For cases where we know two sides and the included angle (SAS) or all three sides (SSS), the Law of Cosines can be used to solve for the remaining sides and/or angles.
If one of the angles of a triangle, say, C, is a right angle, then cos C = 0, and the Law of Cosines reduces to the Pythagorean Theorem, c2 = a2 + b2. Thus the Pythagorean Theorem is a special case of the Law of Cosines.

3. Example 1 - SAS, the Law of Cosines
Two sides of a triangle are 15 and 20. The angle between the two sides measures 124°. Determine the length of the other side and the measure of the other two angles.

4. Example 2 – SSS, the Law of Cosines
The sides of a triangle are a = 5, b = 8, and c = 12. Find the angles of the triangle. Round to 2 decimal places.

5. Navigation: Heading and Bearing
In navigation a direction is often given as a bearing, that is, as an acute angle measured from due north or due south. The bearing N 30 E, for example, indicates a direction that points 30 to the east of due north.

6. Example 3 – Navigation
A pilot sets out from an airport and heads in the direction N 20 E, flying at 200 mi/h. After 1 h, he makes a course correction and heads in the direction N 40 E. Half an hour after that, engine trouble forces him to make an emergency landing.
(a) Find the distance between the airport and his final landing point.
(b) Find the bearing from the airport to his final landing point.

7. Example 4 – Navigation
Ship A departs with a bearing of S 50 W traveling 40 mph. Another ship, ship B, departs at the same time as ship A at a bearing of N 10 W traveling 25 mph.
(a) Find the distance between the two ships 3 hours after departure.
(b) Suppose 3 hours after departure, ship B has encountered engine failure and is stranded. If ship A can maintain a speed of 40 mph, how long would it take ship A to reach the stranded ship B?

8. The Area of a Triangle
Heron’s formula us used to find the area of a triangle from the lengths of its three sides.

9. Example 5 – Area of a Lot
A businessman wishes to buy a triangular lot in a busy downtown location (see Figure 9). The lot frontages on the three adjacent streets are 125, 280, and 315 ft. Find the area of the lot.
Figure 9

10. WebAssign – Question 19
A steep mountain is inclined 74° to the horizontal and rises h = 3400 ft above the surrounding plain. A cable car is to be installed from a point d = 800 ft from the base to the top of the mountain, as shown. Find the shortest length of cable needed. (Round your answer to the nearest foot.)