Law of Cosines

c2 = a2 + b2 – 2abcos(θ)

• Use to find third side of a triangle

• Or to solve for unknown angles

• When c is the hypotenuse of a right triangle we get Pythagoras’ theorem!

Finding a third side length

• When you have 2 lengths and the angle between them.

a = 7

b = 12

θ = 40°

c2 = a2 + b2 – 2abcosθ

= 49 + 144 – 168cos(40°)

= 64.305

c = 8.02

Solving for an angle

• When you have all three side lengths

a = 9

b = 14

θ = ?

c = 11

c2 = a2 + b2 – 2abcosθ

2abcosθ = a2 + b2 – c2

θ = cos-1 ( )a2 + b2 – c2

2ab

θ = cos-1(0.619)θ = 51.75°

Law of Sines

• Use to find side length(s) of a triangle

• Or to solve for any unknown angle(s)

• In a right triangle we get sin(θ) =

a b csin(A) sin(B) sin(C)

= =

opphyp

Finding an unknown side length

• When you have at least one length and two angles.

a = 7

b = ?

A = 40°

B = 110°

sin(A) sin(B) a b=

sin(A) sin(B)b = a·

sin(40)

sin(110)b = 7·

b = 10.23

b = 10.23

Problems1. Use the law of cosines to find the measure of the largest

angle in a 4-5-6 triangle.

2. Use the law of sines to find the shortest side in a 40°-60°-80° triangle whose longest side is 10.0 cm.

3. A triangle has known angles of 37° and 55°. The side between them is 13 cm long. Find the other side lengths.

4. A plane which is 100 miles due West of you moves in a roughly Northerly direction at 400 mph. If after 10 minutes the new distance to the plane is 130 miles, determine the exact heading of the aircraft.