Law of Sines
TheLaw of Sinesis the relationship between the sides and angles of non-right (oblique)triangles. Simply, it states that the ratio of the length of a side of a triangle to the sine of the angle opposite that side is the same for all sides and angles in a given triangle.
在is an oblique triangle with sidesand, then.
To use the Law of Sines you need to know either two angles and one side of the triangle (AAS or ASA) or two sides and an angle opposite one of them (SSA). Notice that for the first two cases we use the same parts that we used to prove congruence of triangles in geometry but in the last case we could not provecongruent trianglesgiven these parts. This is because the remaining pieces could have been different sizes. This is called the ambiguous case and we will discuss it a little later.
Example 1:Given two angles and a non-included side (AAS).
Givenwith,andm. Find the remaining angle and sides.
The third angle of the triangle is
由Law of Sines,
由Properties ofProportions
and
Example 2:Given two angles and an included side (ASA).
Given,andcm. Find the remaining angle and sides.
The third angle of the triangle is:
由Law of Sines,
由Properties of Proportions
and
The Ambiguous Case
If two sides and an angle opposite one of them are given, three possibilities can occur.
(1) No such triangle exists.
(2) Two different triangles exist.
(3) Exactly one triangle exists.
Consider a triangle in which you are givenand. (The altitudefrom vertexto side, by the definition of sines is equal to.)
(1) No such triangle exists ifis acute andoris obtuse and.
(2) Two different triangles exist ifis acute and.
(3) In every other case, exactly one triangle exists.
Example 1:No Solution Exists
Givenand. Find the other angles and side.
Notice that. So it appears that there is no solution. Verify this using the Law of Sines.
This contrasts the fact that the. Therefore, no triangle exists.
Example 2:Two Solutions Exist
Givenand. Find the other angles and side.
therefore, there are two triangles possible.
由Law of Sines,
There are two angles betweenandwhose sine is approximately 0.5833, areand.
Example 3:One Solution Exists
Givenand. Find the other angles and side.
由Law of Sines,
is acute.
由Law of Sines,
If we are given two sides and an included angle of a triangle or if we are givensides of a triangle, we cannot use the Law of Sines because we cannot set up any proportions where enough information is known. In these two cases we must use theLaw of Cosines.