Using Law of Sines and Law of Cosines

When do you use Law of Sines and when do you use Law of Cosines?

Well,

Law of Sines

more

AAS

SSA (ambiguous)

Law of Cosines

more sides given

SAS

SSS

Ambiguous Case: SSA

There are five possible combinations of triangles when you are given two sides and an angle.



In Case 1, you can see that side b >a

In Case 2, b forms a single right angle triangle with c. Here b equals h, h being the height of the triangle, and yields a single right angle triangle.

In Case 3, a>b>h, forming two triangles. Side b is too long to form a single right angle triangle, but yet is also too short to swing out farther than side a which would result in only one triangle. Instead it forms to triangles, one acute triangle and one obtuse triangle.

In Case 4, side b is equal side a, resulting in a single isosceles triangle. Being an isosceles triangle, angle A and angle B are also equal. Side b cannot be placed anywhere else or it would not form a triangle.

In Case 5, side b is > side a. It forms one triangle only, with side b stretching out opposite of a. It cannot be on the other side of a because then it would not form a triangle.

Ok here is how we use the Law of Sines to solve a triangle.

Lets say we are given:
a = 21
b=20

Lets start by solving for
(sinA)/a = (sinB)/b

So we plug in the numbers that we have:

(sin 33)/21 = (sinB)/20
20(0.545)/21=sinB
0.519=sinB
B=31.268°

We have found
180-33-31.268=115.732°

We then can find c with the Law of Cosines:

c(c)=a(a)+b(b)-2abcosC
c(c)=441+400-2(21)(20)cos115.732
c(c)=1205.696
c=34.723