Section 5-8 The Law of

Cosines

A

b c

C a B

Solve ∆ABC ifA= 120⁰, b=9, c=5

=12.3

=12.3

=39.3⁰

Solve ∆ABC ifA= 105⁰, b=12, c=9

A

105⁰ 12 9

C a B

a2 = b2 + c2 - 2bc cos A Law of Cosinesa2 = 122 + 92 - 2(12)(9) cos 105°a2 = 280.9049137a = 16.76021819

So, a = 16.8.

B = 43.6°.

C = 180° - (105° + 43.6°)C = 31.4°

B = 43.6°.

C = 180° - (105° + 43.6°)C = 31.4°

Solve ∆ABC ifA= 105⁰, b=12, c=9

A

105⁰ 12 9

C a B

A triangle ABC has a = 8, b = 9, and c = 7. What is the measure of angle C?

A triangle ABC has a = 7, b = 6, and angle A = 80º. Find the measure of side c.

Two airplanes leave an airport, and the angle between their flight paths is 40º. An hour later, one plane has traveled 300 miles while the other has traveled 200 miles. How far apart are the planes at this time?

To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245 yards. He then turns 110º and walks 270 yards until he arrives at the other end of the lake. Approximately how long is the lake?

To approximate the length of a lake, a surveyor starts at one end of the lake and walks 245 yards. He then turns 110º and walks 270 yards until he arrives at the other end of the lake. Approximately how long is the lake?

• After the hurricane, the small tree in my neighbor’s yard was leaning. To keep it from falling, we nailed a 6-foot strap into the ground 4 feet from the base of the tree. We attached the strap to the tree 3½ feet above the ground. How far from vertical was the tree leaning?

• After the hurricane, the small tree in my neighbor’s yard was leaning. To keep it from falling, we nailed a 6-foot strap into the ground 4 feet from the base of the tree. We attached the strap to the tree 3½ feet above the ground. How far from vertical was the tree leaning?

Hero’s Formula

Find the area of ∆ ABC.