1. Law of Cosines c2 = a2 + b2 – 2abcos(θ) Or to solve for unknown angles
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!

2. Finding a third side length
When you have 2 lengths and the angle between them.
c2 = a2 + b2 – 2abcosθ
= – 168cos(40°)
a = 7 =
c = 8.02
θ = 40°
b = 12

3. Solving for an angle When you have all three side lengths
c2 = a2 + b2 – 2abcosθ
2abcosθ = a2 + b2 – c2
a = 9
c = 11
θ = cos-1 ( )
a2 + b2 – c2
2ab
θ = 51.75°
θ = ?
θ = cos-1(0.619)
b = 14

4. Law of Sines a b c = = sin(A) sin(B) sin(C)
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) =
opp
hyp

5. Finding an unknown side length
When you have at least one length and two angles.
sin(A) sin(B)
a b =
sin(A)
sin(B)
b = a·
sin(40)
sin(110)
b = 7·
B = 110°
a = 7
b = 10.23
A = 40°
b = ?
b = 10.23

6. Problems
Use the law of cosines to find the measure of the largest angle in a triangle.
Use the law of sines to find the shortest side in a 40°-60°-80° triangle whose longest side is 10.0 cm.
A triangle has known angles of 37° and 55°. The side between them is 13 cm long. Find the other side lengths.
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.