Question

PLEASE HELP
6.02B

Respond to the following prompt in a word processing document.

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. To aid in your explanation, you may refer to specific problems from the text.

Your response must include:
A discussion of
The law of cosines
The law of sines
The ambiguous case (law of sines)
General guidelines in your own words that can be applied to any triangle.

Answer 1

Law of cosines :

The law of cosines establishes:

$c ^ 2 = a ^ 2 + b ^ 2 - 2*a*b*cosC.$

general guidelines:

The law of cosines is used to find the missing parts of an oblique triangle (not rectangle) when either the two-sided measurements and the included angle measure are known (SAS) or the lengths of the three sides (SSS) are known.

Law of the sines:

In ΔABC is an oblique triangle with sides a, b, and c, then:

$\frac{a}{sinA} =\frac{b}{sinB} =\frac{c}{sinC}$

The law of the sines is the relation between the sides and angles of triangles not rectangles (obliques). It simply states that the ratio of the length of one side of a triangle to the sine of the angle opposite to that side is equal for all sides and angles in a given triangle.

General guidelines:

To use the law of the sines you need to know either two angles and one side of the triangle (AAS or ASA) or two sides and an opposite angle of one of them (SSA).

The ambiguous case :

If two sides and an angle opposite one of them is given, three possibilities may occur.

(1) The triangle does not exist.

(2) Two different triangles exist.

(3) Exactly a triangle exists.

If we are given two sides and an included angle of a triangle or if we are given 3 sides of a triangle, we can not use the law of the sines because we can not establish any proportion where sufficient information is known. In these two cases we must use the law of cosines