Pg. 435 Homework• Pg. 443 #1 – 10 all, 13 – 18 all

• #3 ɣ = 110°, a = 12.86, c = 18.79 #4 No triangle possible• #5 α = 90°, ɣ = 60°, c = 10.39• #6 b = 4.61, c = 4.84, ɣ = 68°• #7 a = 3.88, c = 6.61, ɣ = 68°• #8 α = 41.62°, ɣ = 53.38°, c = 4.83• #10 α = 29.51°, ɣ = 112.49°, c = 30.01• #12 α = 49.51°, ɣ = 14.49°, c = 3.62• #13 No triangle possible #14 No triangle possible• #15 α = 22.06°, ɣ = 5.94°, c = 2.20• #18 a)54.60 ft b) 51.93 ft apart• #21 0.72 miles high

8.2 Law of Cosines

Definition• Law of Sines is best for

when you have two angles and one side or when two sides and a non-included angle are given.

• Law of Cosines is best when you have two sides and the included angle (Law of Sines does not apply here!)

Law of Cosines• For any triangle ABC,

labeled in the usual way

• Solve triangle ABC ifa = 5, b = 3, ɣ = 35°.

• Solve triangle ABC ifa = 9, b = 7, c = 5.

2 2 2

2 2 2

2 2 2

2 cos

2 cos

2 cos

a b c bc

b a c ac

c a b ab

8.2 Law of Cosines

Solving an Oblique TriangleParts Given Number of

Possible Triangles1. Three Angles (sum equals 180°)

Infinitely Many

2. Two Angles (sum less than 180°) and One Side

One

3. One Angle, One Side

Zero, One or Two

4. Three Sides (sum of any two greater than the third)

One

Area of Triangles• Let ABC be a triangle

labeled in the usual way. Then the area A of the triangle is given by:

1 sin21 sin21 sin2

A bc

ac

ab

8.2 Law of Cosines

Heron’s Area Formula• Let ABC be a triangle

labeled in the usual way. Then the area A of the triangle is given by:

where is one half the perimeter, or the semiperimeter.

Examples:• Use the best method to find

the area.• Find the area of triangle

ABC if a = 8, b = 5, ɣ = 52°.• Find the area of triangle

ABC if a = 9, b = 7, c = 5. A s s a s b s c

12s a b c