6.1 Law of Sines

An Oblique Triangle is any triangle that does not contain a right angle. 

Pictures of Derivation- used from http://www.regentsprep.org/Regents/math/algtrig/ATT12/derivelawofsines.htm

The law of sines can be used to solve an oblique triangle in two different situations. The first is when given two angles and one side. The other situation is when given to sides and one angle. 

AAS

C=102.3° B=28.7° b=27.4 ft

First, solve for the missing angle. To do this, subtract the two known angles from the total degrees in a triangle, 180°. 

A=180°-B-C

A=180°-102.3°-28.7°

A=49°

Now, we can use the law of sines to solve for the two missing sides.

After solving both of these equations, it can be found that a=43.06 ft and c=55.75 ft.

ASA

A=43° B=98° c=22 ft

First, solve for the missing angle just like the last example. 

C=180°-A-B

C=180°-43°-98°

C= 39°

Next, use the law of sines to solve for the two missing sides.

Once you solve both of these equations, it can be found that a=23.84 ft and b=34.62 ft.

SSA

When given two sides and an opposite angle, a unique triangle is not always determined. There are three different possibilities of a SSA triangle- a single solution, no solutions, or two solutions.

a=12m b=31m A=20.5°

When this equation is simplified, it is found that sinB=.9047. However, there are two angles that have a sine of .9047 that are less than 180°. Thus, there are two possible triangles have these sides and this angle in it. REMEMBER, your calculator will only show one of the solutions because they restrict the domain of the inverse sine function.   

Thus, B is either 64.8° or 115.2°. After solving for this angle, it becomes easy to finish solving each triangle. To find C simply subtract angles A and B from 180°. Then, to solve for the remaining side, repeat the law of sines.

Area using Law of Sines

The area of an oblique triangle can be solved using a formula, which was derived using the law of sines. The formula is:

Heres an example of how to use it:

A triangle has sides a=90m b=52m and angle C=102°. What is the area?