Question

Describe, in detail, when to use the law of cosines, the law of sines, and the law of sines with the ambiguous case.

Answer 1

you can memorize when to use each, but it is silly.
as the law of sines is easiest you should always use that if you can. in order to use
\[\frac{\sin(A)}{a}=\frac{\sin(B)}{b}\] what you need is three out of the four numbers. if you do not have three of those four, you cannot use it.

so for example if you have only 3 sides and no angles you can't use it, or if you have two sides and an included angle you cannot use it, and have to use the law of cosines

Answer 2

if your given two angles or three angle and one side then use sine law.
if you are given only sides then use cosine law, to find angles

Answer 3

how can i explain this in an essay form??

Answer 4

i think that english is really up to you yes? i wrote off the top of my head. if you understand what i wrote, put it in your own words.

btw i started with math exactly so i would not have to write "essays"
just say it in plain english, and if your spelling and punctuation is good your teacher will be happy

Answer 5

The Law of Cosines is best suited for the situation when three sides of a triangle are known but no angle. Also it is convenient when two sides and the included angle are known.
The Law of Cosines is useful for an ambiguous case because it returns a negative cosine is the angle is obtuse unlike the Law of Sines.
The Law of Sines is simpler and easier to use. However care must be taken in the case that the given information is SSA because that can lead to ambiguity. If the given angle is 45 degrees and the sine returned is 1/2, there are two angles less than 180 degrees whose sine is 1/2. They are 30 degrees and 150 degrees. In this case one would choose 30 degrees because to choose 150 degrees would result in a sum greater than 180 degrees. But if the given angle is 43.1 degrees and the sine returned is .91225 then the two possible angles are 65.8 degrees and 114.2 degrees. And so there are two triangles that satisfy the given conditions. If angle A is given to be 42 degrees and a = 70 and b = 122, applying the Law of Sines results in sin B = 1.17. Since the sine of an angle is never greater than 1, there is no triangle possible.