Question

Describe, in detail, when to use the law of cosines, the law of sines, and the law of sines with the ambiguous case. Provide general guidelines, in your own words, for each law that can be applied to any triangle situation with which you are presented.

Answer 1

The Law of Cosines is
c² = a² + b² - 2abcosΘ.
It has four variables; a, b, c and Θ.
Whenever you have a triangle with three of these variables known, use the Law of Cosines to calculate the unknown variable.

Answer 2

The Law of Sines is
a/sinA = b/sinB = c/sinC
Although it has six variables, when you use it, you only need four.
You can use a/sinA = b/sinB, or a/sinA = c/sinC, or b/sinB = c/sinC.
Whenever you have a triangle with three of those four variables known, you use the Law of Sinces to calculate the unknown variable.

Answer 3

Because the sine of an angle is equal to the sine of the supplement of that angle, that is an ambiguity. Therefore, when you use the Law of Sines, you have two values for each sine. One will be an acute angle and one will be an obtuse angle. These two angles will be supplements of each other. In other words, the sum of these two angles = 180º.