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

Law of Cosines In ∆ABC, or a c b B A C

* 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.

Examples a. b. 5 x 6 35° 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 C D B

Find the area of the quadrilateral: Example °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 B A 44 2.5  18,000 ft 23,000 ft

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