9.4 – The Law of Cosines

Essential Question: How and when do you use the Law of Cosines?

Law of Cosines

• In ∆ABC,

•

• or

• or

a c

b

B

AC

Cabbac cos2222

Baccab cos2222

Abccba cos2222

• * Used in SAS (included angle):

(side opp. <)2 = (side adj. to <)2 + (other side adj. to <)2 –

2 (adj. side) (other adj. side) cos <

Used when finding a side.

• *Or used in SSS:

• Used when finding an angle.

.).)((2

.)(.)(.)(cos

222

adjadj

oppadjadj

Examples

a.

b.

5x

635°

5

10

x

115°

• Solve each triangle. Give measurements to the nearest tenth.

a) a = 8, b = 5, <C = 60°

b) p = 3, q = 8, <R = 50°

Examples

• A Δ has sides of lengths 6, 12, and 15.

a. Find the measure of the smallest <.

b. Find the length of the median to the longest side.

Example

A parallelogram has diagonals of lengths 20 cm and 12 cm. If the diagonals intersect to form a 60 angle, find the perimeter of the parallelogram.

Examples

Find AD, if AB = 8, BD = 7, DC = 5, AC= 10.

Example

A

CD

B

Find the area of the quadrilateral:

Example

1210

10

108° 132°

Two airplanes, at points A and B in the diagram below, have elevations of 23,000 feet and 18,000 feet respectively. Both are flying east toward an airport control tower at T. From T, the angle of elevation of the airplane at A is 4°, and the angle of elevation of the airplane at B is 2.5°. How far apart (in miles) are the airplanes?

Example

x

T

BA

42.518,000 ft

23,000 ft

How do we know when we should use the law of sines or the law of

cosines?