Law of Cosines

UUUGGGGHHHH this is my THIRD time writing this blog... I must be severely stupid to refresh the page TWICE...

Law of Cosines is used the find missing parts of a triangle when you aren't given opposite angles and sides. In other words, when you can't use Law of Sines or SOHCAHTOA. Law of Cosines is a last resort.

Formula:
leg^2 = (adjacent leg)^2 + (other adjacent leg)^2 - 2(adjacent leg)(other adjacent leg)cos(angle between)

Ex. 1
b = 5, a = 7, and angle C = 40°. Solve the triangle. (This is the triangle above.)

Solving for c:
Plug into formula:
c^2 = 5^2 + 7^2 - 2(5)(7)cos(40)

Take square root:
c = sqrt[5^2 + 7^2 - 2(5)(7)cos(40)]

Plug into calculator:
c ≈ 4.514

Solving for angle A:
Plug into formula:
7^2 = 4.514^2 + 5^2 - 2(4.514)(5)cosA

Solve for cosA:
7^2 -4.514^2 -5^2 = -2(4.514)(5)cosA
(7^2 -4.514^2 -5^2) / (-2(4.514)(5)) = cosA
A = cos^-1[(7^2 -4.514^2 -5^2) / (-2(4.514)(5))]

Plug into calculator:
A ≈ 94.605°
≈ 83.395°

To solve for the angle of B, you can just subtract the angles from 180.
Scenario 1: 180 - 94.605 - 40 ≈ 45.395°

Scenario 2: 180 - 83.395 - 40 ≈ 56.605°


Scenario 1:
a = 7
b = 5
c = 4.514
angle A ≈ 94.605°
angle B ≈ 45.395°
angle C = 40°

Scenario 2:
a = 7
b = 5
c = 4.514
angle A ≈ 83.395°
angle B ≈ 56.605°
angle C = 40°